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+page_content='CODES AND MODULAR CURVES  by  Alain Couvreur  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — These lecture notes have been written for a course at the Algebraic Coding Theory (ACT)  summer school 2022 that took place in the university of Zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The objective of the course propose an  in–depth presentation of the proof of one of the most striking results of coding theory: Tsfasman Vlăduţ  Zink Theorem, which asserts that for some prime power q, there exist sequences of codes over Fq whose  asymptotic parameters beat random codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Contents  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' 32  Introduction  Algebraic Geometry (AG) codes is a particularly exciting topic lying at the intersection between  number theory, algebraic geometry and coding theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The birth of this research area dates back to the  early 80’s with the introduction by Goppa [Gop81] of a new family of codes obtained by evaluating  residues of some differential forms on a given curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Quickly after, Tsfasman, Vlăduţ, Zink [TVZ82] and  independently Ihara [Iha81] proved the existence of sequences of modular curves and Shimura curves  having an excellent asymptotic ratio number of points v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An immediate but extremely striking  corollary is the existence of sequences of codes beating the Gilbert Varshamov bound, in short: codes  better than random codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This remarkable and totally unexpected result turned out to be the first stone  of the development of a whole theory: that of AG codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Surprisingly, a very comparable breakthrough  happened in graph theory the late 80’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, in 1988, Lubotsky, Philips and Sarnak [LPS88] and  independently Margulis [Mar88] used Cayley graphs on quotients of SL2(Z) to prove the existence of a  family of graphs whose girth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' the length of their shortest cycle, exceeds the girth obtained with the  probabilistic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In both situations, coding theory and graph theory, the use of elegant algebraic  structures unexpectedly beat random constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The objective of this lecture is to present in an (almost) self-contained presentation, the beginning of  this wonderful story: the original proof of Tsfasman, Vlăduţ and Zink Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It should be mentioned  that in 1995, Garcia and Stichtenoth [GS95] proposed another and somehow more explicit approach  to design sequence of curves (actually function fields but the two objects are equivalent) reaching the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='03569v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='NT]  9 Jan 2023  2  ALAIN COUVREUR  so-called Drinfeld–Vlăduţ [VD83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It could be considered as strange to present the original proof which  turns out to be much more complicated than Garcia and Stichtenoth’s one but there are some reasonable  motivations for that:  Tsfasman, Vlăduţ and Zink’s proof testifies from the richness of the theory of algebraic geometry  codes, with a proof involving deep results from algebraic geometry and number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This original proof is frequently cited while few references give a complete presentation of it and  (in my personal opinion), none of the papers of Tsfasman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' and Ihara provide an enough  detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In both articles, the proof is made of less than ten lines hiding a huge amount of  prerequisites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, I wished to give that lecture, because this proof is beautiful and elegant and even if I am  not among the mathematicians who do maths pour la beauté de la chose(1) it is sometimes pleasant  to take the time to appreciate the elegance of some development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Outline of these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — We start in Section 1 with bases on linear codes and their asymptotic  behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Section 2 gives an introduction to algebraic curves by providing the necessary material in  algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Section 3 introduces algebraic geometry codes and states the main result: Tsfasman–  Vlăduţ–Zink Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The remainder of the notes are dedicated to the proof of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Sections 4  and 5 provide further material on elliptic and modular curves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Section 6 concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — First, I would like to thank Gianira Alfarano, Karan Khaturia, Alessandro  Neri, Violetta Weger, the organisers of the Algebraic Coding Theory Summer School(2) 2022 who gave me  the motivation to type-write old hand-written notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' I would probably never have found the time to do  it if they did not ask me for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Several colleagues spent time to carefully read these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In particular, I  express a deep gratitude to Elena Berardini, Maxime Bombar, Grégoire Lecerf, Jade Nardi, Christophe  Ritzenthaler, Joachim Rosenthal and Gilles Zémor for their relevant comments on the preliminary version  of the notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The author is funded by the french Agence nationale de la recherche for the collaborative project  ANR-21-CE39-0009-BARRACUDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Linear Codes  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the sequel we are interested in linear q–ary codes, which are linear subspaces of Fn  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  What makes the study hard, but also deeply interesting is that we are not only considering elementary  objects such as finite dimensional vector spaces but spaces endowed with a metric: the Hamming metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The Hamming distance between two vectors x, y ∈ Fn  q is denoted by  dH(x, y)  def  = ♯{i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , n} | xi ̸= yi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The Hamming weight of a vector is its Hamming distance to the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  ∀x ∈ Fn  q ,  wH(x)  def  = dH(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Unless otherwise specified, a code will denote a linear subspace C ⊆ Fn  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The  vectors of C are usually referred to as codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The dimension of C regarded as an Fq–vector space is  always denoted by k and its minimum distance denoted by d is defined as  d  def  = min  x,y∈C  x̸=y  {dH(x, y)} =  min  c∈C\\{0} {wH(c)} ,  where the last equality is a consequence of the linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The parameters of a code C ⊆ Fn  q refer to the  triple n, k, d and is usually denoted as [n, k, d]q, where the cardinality q of the base field is recalled in  (1)Litterally : “for the beauty of the thing”  (2)https://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='ch/act/  CODES AND MODULAR CURVES  3  subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally, one can also be interested in the rate and relative distance of a code, respectively  defined and denoted as follows:  R  def  = k  n  and  δ  def  = d  n·  A longstanding problem in coding theory is which kind of triples of parameters [n, k, d] can be achieved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A code will be considered as “good” if both k and d are as close as possible to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  However, many  upper bounds exist, the most elementary one being the Singleton bound saying that for any code with  parameters [n, k, d]q we have  (1)  k + d ⩽ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The rationale behind this question is that both k and d quantify some feature of linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Suppose  we are given a transmission channel, that can be either a wire or a wireless communication for instance  an exchange between electronic devices like between a computer and a WiFi antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The rate is nothing  but the ratio of information divided by the quantity of data which is actually sent across the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Hence, the rate R = k/n quantifies the efficiency of encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  On the other hand, the minimum distance quantifies how far are words from each other and hence the  theoretical ability to recover an original message from a corrupted codeword(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, suppose that our objective is to correct errors from a given channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider for instance the  q–ary symmetric channel with parameter p ∈ [0, 1 − 1  q] which takes as input a vector c ∈ Fn  q and outputs  the vector c + e where e = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , en) and the ei’s are independent random variables over Fq taking  value 0 with probability 1 − p and any other value in Fq \\ {0} with probability  p  q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The average weight  of our error vector satisfies  E(wH(e)) = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  However, for small values of n, deviations may happen and it is possible that our input vector c is  corrupted by much more than ⌊pn⌋ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, it is relevant to consider large values of n for which  the law of large numbers will assert us that the weight of the error will be close to its expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This last discussion motivates the search of sequences of codes (Cs)s∈N with parameters [ns, ks, ds]  where  lim  s→+∞ ns = +∞  and  lim  s→+∞  ks  ns  = R  lim  s→+∞  ds  ns  = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Usually in the literature, the sequences (ks/ns)s and (ds/ns)s are not supposed to  converge and lim sup’s are used instead of actual limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In this setting, the question of the achievable pairs (δ, R) ∈ [0, 1] × [0, 1] remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Some bounds  are known:  Singleton bound immediately entails that R + δ ⩽ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A more precise bound called Plotkin bound entails that R + δ ⩽ 1 − 1  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' See for instance [Cou16,  Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 4]  A principle that “constructing bad codes from good ones is always possible” permits to prove that  give an achievable pair (δ, R) any pair (δ′, R′) with δ′ ⩽ δ and R′ ⩽ R is achievable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove this last assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  More precisely, it has been proved by Manin [VM84], that the frontier between the subdomain  of [0, 1] × [0, 1] of achievable pairs (δ, R) and the non achievable ones is the graph of a continuous  function R = αq(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, if proving the existence and the continuity of this function αq is not  very hard, having an explicit description of it remains a widely open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An upper bound for  αq is given by the minimum of all the known upper bounds on the achievable pairs (δ, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (3)Here we do not introduce any consideration about practical algorithms to correct errors  4  ALAIN COUVREUR  On the other hand a famous result on the average behaviour of random codes referred to as the  Gilbert–Varshamov bound asserts that for a random code(4) C ⊆ Fn  q with fixed rate R, then for any  ε > 0 the probability that the relative distance δ of C satisfies  R ∈ [1 − Hq(δ) − ε, 1 − Hq(δ) + ε],  goes to 1 when n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The function Hq(·) is the q–ary entropy function defined as  Hq :  �  �  �  [0, 1]  −→  R  x  �−→  � − logq(q − 1) − x logq(x) − (1 − x) logq(1 − x)  if  x ̸= 0, 1  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In short, the pair (δ, R) for a random sequence satisfies R = 1 − Hq(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In summary, the unknown function δ �→ αq(δ) whose graph is the frontier of the domain of achievable  pairs (δ, R) is known to be continuous, to be bounded from below by the Gilbert–Varshamov bound  δ �→ 1 − Hq(δ) and bounded from above by the min of all known upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For a long time, it has  been supposed that Gilbert Varshamov bound was optimal and that somehow, no family of codes could  asymptotically beat random codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A breakthrough is due to Tsfasman, Vlăduţ and Zink [TVZ82] who  showed that the asymptotic Gilbert Varshamov bound is not always optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' More precisely, they proved  the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let q = p2 where p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then for any R ∈ [0, 1], there exists a sequence  of codes whose length goes to infinity and whose asymptotic parameters (δ, R) satisfy  R + δ ⩾ 1 −  1  p − 1·  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Actually, the result holds for any q = p2m where p is prime and m ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Actually, the result on codes is the corollary of a statement on the existence of a sequence  of algebraic curves with specific properties (see further Theorem 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This statement on curves has proved  by Tsfasman, Vlăduţ and Zink in [TVZ82] and independently by Ihara in [Iha81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, Ihara did  not rely this result with coding theory while Tsfasman et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  It turns out that, as illustrated by Figures 1 and 2, for q ⩾ 49, such codes beat Gilbert Varshamov  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These codes, are actually far from being random and are constructed using elegant techniques  from number theory and algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The objective of these notes is to outline a proof of this  incredible result, which is probably one of the major breakthroughs of coding theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic curves  The objective of this section is not to provide an in depth lecture of algebraic geometry but only to give  the minimal prerequisites in algebraic geometry to understand the sequel of these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In particular,  here most of the proofs will be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' I encourage any reader who feels comfortable with algebraic  geometry and for whom reading Harsthorne’s book [Har77] is not harder than reading Harry Potter to  skip this section for two reasons:  she/he will not learn anything in it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  for a reader who feels comfortable with the language of schemes, the contents of this section could  appear to be dirty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  If you wish further details on algebraic geometry, I can encourage the following readings depending from  your knowledge on the topic:  Walker’s book [Wal00] is an excellent first reading if you do not know anything about algebraic  geometry and algebraic geometry codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (4)This can be formalised as follows, consider the set of all codes of length n and dimension Rn in Fn  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This set is finite,  and let C be a random variable uniformly distributed over this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9  1  R  δ  GV bound  TVZ bound  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The TVZ bound for q = 49  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='9  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9  1  R  δ  GV bound  TVZ bound  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The TVZ bound for q = 121  If you do not like geometry, Stichtenoth’s book [Sti09] proposes an excellent introduction to alge-  braic geometry codes from a purely arithmetic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It provides in particular a different  proof of the Tsfasman–Vlăduţ–Zink theorem based on so–called recursive towers of function fields  which excludes any geometric consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  6  ALAIN COUVREUR  A more advanced presentation on algebraic geometry codes appears in Tsfasman Vlăduţ and Nogin’s  book [TVN07] and Stepanov [Ste99] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, the reader interested in discovering algebraic geometry out of the context of algebraic coding  theory is encouraged to look (for instance) at the books [Ful89, Sha94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Lorenzini’s book [Lor96]  can be an excellent reading either if you wish a better focus on the arithmetic side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Plane curves and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let K be a perfect field(5) and K be its algebraic closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We  denote by A2(K) and P2(K) respectively the affine and projective planes over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An affine plane curve  X over K is the vanishing locus in A2(K) of a nonzero two variables polynomial f(x, y) ∈ K[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Similarly, a projective plane curve is the vanishing locus in P2(K) of a nonzero homogeneous polynomial  F(X, Y, Z) ∈ K[X, Y, Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Recall that the projective plane P2(K) is the set of vectorial lines of K  3 or  equivalently is the quotient set  P2(K)  def  = (K  3 \\ {0})/K  ×,  and its elements are represented as triples (u : v : w) with the equivalence relation (u : v : w) ∼ (a : b : c)  if there exists λ ∈ K  × such that u = λa, v = λb and w = λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Such an affine (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' projective) curve is said to be irreducible if f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' F) is an irreducible polynomial  in K[x, y] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' K[X, Y, Z]) and absolutely irreducible if f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' F) is irreducible when regarded as an  element of K[x, y] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' K[X, Y, Z]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Suppose K = Q and consider the affine curve X with equation x2 − 2y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This  curve is irreducible but not absolutely irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, over Q, the equation of the curve factorizes as  (x−  √  2y)(x+  √  2y) = 0 and this factorisation is not defined over Q: the polynomial x2−2y2 is irreducible  over Q but not over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Geometrically speaking, X is the union of the two lines with respective equations  x −  √  2y = 0 and x +  √  2y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These lines are not defined over Q but their union is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given an affine irreducible plane curve X , the quotient ring K[x, y]/(f) is integral and its field of  fractions Frac(K[x, y]/(f)) is well–defined and referred to as the function field of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the projective  setting, the function field can also be defined as the field of fractions A(X,Y,Z)  B(X,Y,Z) where A, B are homogeneous  polynomials of the same degree with B is not divisible by F and with the relation:  A(X, Y, Z)  B(X, Y, Z) = C(X, Y, Z)  D(X, Y, Z)  if  F divides (AD − BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  For an affine curve X , elements of K[x, y]/(f) can be understood as restrictions of polynomial functions  to the curve X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, considering two polynomials a(x, y), b(x, y) ∈ K[x, y] regarded as functions  A2(K) → K, one can consider their restrictions to X and a well–known result usually called Hilbert’s  Nullstellensatz (see for instance [Ful89, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='7]) asserts that their restrictions to X are the same if and  only if f divides a − b and hence if and only if they are congruent modulo the ideal spanned by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In the projective setting, a homogeneous polynomial cannot be interpreted as a function P2(K) → K  since an element of P2(K) is described by a triple (u : v : w) but also by any other triple (λu : λv : λw)  for any λ ∈ K  ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence, given a non constant homogeneous polynomial P ∈ K[X, Y, Z] of degree d > 0,  the evaluation cannot make sense since P(λu, λv, λw) = λdP(u, v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Note however that, for such a  polynomial, vanishing at a point is a well–defined notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, the evaluation of a fraction P/Q of  two homogeneous polynomials with the same degree makes sense since  P(λu, λv, λw)  Q(λu, λv, λw) = λdP(u, v, w)  λdQ(u, v, w) = P(u, v, w)  Q(u, v, w)·  This is the reason why we introduce these objects as the good definition of functions on a projective  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Note that we are juggling with K and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Here it is crucial no notice that the curve  is defined as a set of points with coordinates in K, while functions, should be rational functions with  coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On one hand, the function field is defined over K and describes the arithmetic of the  (5)In the sequel the fields of interest will be either C or finite fields Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  7  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On the other hand, when describing a curve as a set of points, considering only the points with  coordinates in K would be too poor: think for instance about the case where K is a finite field, in this  situation the set of points with coordinates in K is finite and might actually be empty!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Then, very  different equations may provide the same set of points with coordinates in K while the sets of points over  K will be very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This explains the rationale behind considering the points with coordinates in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Note that when speaking about functions, these objects may not be defined everywhere  on the curve and may have some poles somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These objects can be understood as the algebraic  geometric counterpart of meromorphic functions in complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — It is well–known that the projective plane can be covered by affine planes sometimes  called affine charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed one can embed the affine plane into P2 as:  �  A2(K)  −→  P2(K)  (x, y)  �−→  (x : y : 1)  or  �  A2(K)  −→  P2(K)  (x, y)  �−→  (x : 1 : y)  or  �  A2(K)  −→  P2(K)  (x, y)  �−→  (1 : x : y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The images of these three embeddings cover the full projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence, given a projective curve,  one can consider the restriction of the curve on the image of one of the above embeddings and get an  affine curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Practically, starting with a projective curve with equation F(X, Y, Z) = 0 one can consider  for instance the affine curve with equation F(x, y, 1) = 0 but also those with equations F(x, 1, y) = 0  or F(1, x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence, one can deduce affine curves (affine charts) from a given projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On  the other hand, starting from an affine curve X with equation f(x, y) = 0 the homogeneous polynomial  F(X, Y, Z) of degree deg f such that f(x, y) = F(x, y, 1) (such a homogeneization is unique, details are  left to the reader) is the equation of a curve sometimes referred to as the projective closure of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A crucial fact is that a curve and its projective closure share a common object : their function field  remains the very same one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A point of X is an element (a, b) ∈ A2(K) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' (u : v : w) ∈ P2(K)) such that  f(a, b) = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' F(u, v, w) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A point is said to be a rational point or a K–point if its coordinates  all lie in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' More generally, given an extension L/K, one can define the notions of L–points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The  set of K–points or L–points of X respectively denoted by X (K) and X (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' One of topics of interest  for us in the sequel is the case K = Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this situation, one sees easily that X (Fq) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, it  is a subset of A2(Fq) or P2(Fq) which are both finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On the other hand X has been defined as a  set of K–points that we sometimes call the geometric points in the sequel, hence we can also denote it as  X (K) when we wish to emphasize that we are interested in any possible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given an affine (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' projective) curve X defined by the equation f(x, y) = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' F(X, Y, Z) = 0)  over a field K, a point P ∈ X (K) with coordinates (xP , yP ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' (uP : vP : wP )) is said to be singular  if  ∂f  ∂x(xP , yP ) = ∂f  ∂y (xP , yP ) = 0  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  ∂F  ∂X (uP , vP , wP ) = ∂F  ∂Y (uP , vP , wP ) = ∂F  ∂Z (uP , vP , wP ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A non singular point is said to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A curve without singular points is said to be regular or smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  On the other hand a curve having at least one singular point is said to be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It can be proved that  the set of singular points of a curve is always finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  From now on, unless otherwise specified, any curve is smooth projective and absolutely irre-  ducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Galois action on points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Recall that, for the sake of simplicity, we restrict the definitions to  the case where the base field K is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This is not a strong restriction for the subsequent purpose  where K will always be either finite or of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given a curve X defined over K, any point P ∈ X (K) has coordinates (xP , yP ) (or (uP : vP : wP ) in  the projective setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These coordinates being in K while X is defined by polynomial equations with  coefficients in K, there is a natural action of Gal(K/K) on points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Note that the coordinates of P  8  ALAIN COUVREUR  are algebraic over K and hence generate a finite extension of K usually denoted K(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, even if  Gal(K/K) may be a complicated object (a profinite group), P is stabilized by Gal(K/K(P)) and hence  the orbit of P is a finite set which is nothing but the orbit of P under the action of the finite group  Gal(K(P)′/K), where K(P)′ is the Galois closure of K(P) over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let K be a perfect field, a closed point of a curve X defined over K is the orbit of a  geometric point P ∈ X (K) under the action of the absolute Galois group Gal(K/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The number of elements in such an orbit is referred to as the degree of the closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It is also  the extension degree [K(P) : K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A rational point is always closed since it is fixed by any element of the  absolute Galois group and hence it equals to its own orbit under this group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — If you prefer the language of number theory, closed points are nothing but the geometric  analogue of the places of the function field K(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Consider the case K = Q and the affine curve C with equation x2 + y2 − 1 = 0  (a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The point with coordinates (1, 0) is a rational point of X , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' an element of C (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The  complex point (2,  √  3i), (where i2 = −1), is a geometric point of C , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' an element of C (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally,  {(2, i  √  3), (2, −i  √  3)} is a closed point of degree 2 of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Maps between curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — As usually in algebra, once structures have been introduced: for  instance groups, rings, modules, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', one introduces morphisms between these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the case of  curves, we are interested in two kinds of maps referred to as morphisms and rational maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A rational  map between two affine (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' projective) curves X , Y contained in A2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' P2) is a map:  ϕ :  �  X  ���  Y  (x, y)  �−→  (ϕ1(x, y), ϕ2(x, y))  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  ψ :  �  X  ���  Y  (u : v : w)  �−→  (ψ1(u, v, w), ψ2(u, v, w), ψ3(u, v, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  where φ1, φ2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' ψ1, ψ2, ψ3) are elements of K(X ) (and, in the projective setting, at least one of the  three functions ψ1, ψ2, ψ3 is nonzero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The dashed arrow ��� is here to emphasize the fact that this  map is not defined at every point but only on a subset(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For affine curves, at any point where ϕ1, ϕ2  have no pole, the map is defined and said to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For projective curves, at any point P where for  some η ∈ K(X )×, ηψ1, ηψ2, ηψ3 have no pole at P and are not simultaneously vanishing, the map ψ is  well–defined and said to be regular at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A rational map between two curves X ��� Y is said to be  regular if it is regular at any point of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A rational map ϕ : X ��� Y induces a field extension the other way around K(Y ) �→ K(X ) which  is defined as follows:  h ∈ K(Y ) �−→ h ◦ ϕ ∈ K(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The degree of ϕ is defined as the degree of this field extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Back to example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The map  (2)  �  C  ���  P1  (x, y)  �−→  (x : 1)  is a rational map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It is also possible to construct a rational map P1 → C as  (3)  �  P1  ���  C  (u : v)  �−→  �  v2−u2  u2+v2 ,  2uv  u2+v2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that these two maps are not inverses to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, the following statements are well–known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Their proofs are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (6)This subset turns out to be dense for a suitable topology called Zariski topology  CODES AND MODULAR CURVES  9  Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let h : X  → Y  be a rational map between two smooth projective absolutely  irreducible curves X , Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (i) if X is smooth, then h is regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (ii) if h is non constant, then it is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Recall that a local ring is a ring having a unique maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The term local  comes precisely from the fact that many such rings may be understood as rings of functions characterized  by a local property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For instance, given an affine curve X and a rational point P with coordinates  (xP , yP ), the ring OX ,P defined as the subring of K(X ) of functions which are regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' have no  pole) at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely  OX ,P  def  =  �a(x, y)  b(x, y) ∈ K(X )  ��� b(xP , yP ) ̸= 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  One can prove that this ring is a local one whose maximal ideal is the ideal:  mX ,P  def  =  �a(x, y)  b(x, y) ∈ K(X )  ��� b(xP , yP ) ̸= 0 and a(xP , yP ) = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  When the point P is smooth, the ring OX ,P is known to be a discrete valuation ring, which means  that the maximal ideal mX ,P is principal and that, given a generator t of this maximal ideal, for any  nonzero element a ∈ OX ,P , there exists a non negative integer n and an element ϕ ∈ O×  X ,P such that  a = ϕtn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such a generator t of mX ,P is called a local parameter (or sometimes a uniformising parameter)  at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, the integer n does not depend on the choice of the generator t and is referred to as the  valuation of a at P and denoted as vP (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, one can easily prove that K(X ) is nothing but the field  of fractions of OX ,P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, any function h ∈ K(X ) can be written as h = h1  h2 ∈ K(X ) \\ {0}, where  h1, h2 ∈ OX ,P and the valuation of h at P will be defined as  vP (h) = vP (h1) − vP (h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In summary, we introduced a map  vP : K(X ) \\ {0} → Z  and this map is known to satisfy the following properties,  ∀a, b ∈ K(X ) \\ {0}, vP (ab) = vP (a) + vP (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  ∀a, b ∈ K(X ) \\ {0}, vP (a + b) ⩾ min{vP (a), vP (b)} and equality holds when vP (a) ̸= vP (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, it should be emphasized that, even if we defined the notion at a rational point, one can actually  extend the notion to any geometric point by replacing K(X ) by K(X ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' the field of rational functions  on X with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the valuation may be defined at any possible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A fundamental object when studying the geometry and arithmetic of a curve is  divisors which somehow are the curve/function fields counterpart of fractional ideals in the theory of  number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given a smooth curve X over a perfect field K, a (geometric) divisor is a formal Z–linear combination  of geometric points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A divisor is said to be rational if it is globally invariant under the action of  Gal(K/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Equivalently, it is a formal sum of closed points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Hence a divisor G on X can be represented as  (4)  G = n1P1 + · · · + nrPr,  where the ni’s are integers and the Pi’s are geometric points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The set {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pr} is referred to as  the support of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The divisor is rational if for any i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , r} such that Pi, Pj are in the same orbit  under the action of Gal(K/K), then ni = nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — We emphasize that a sum of rational points yields a rational divisor but the converse  is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A rational divisor may be a sum of non rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' See the subsequent Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Back to the curve of Example 12 defined over Q with equation x2+y2−1 = 0, consider  the points P = ( 1  2,  √  3  2 ), P ′ = ( 1  2, −  √  3  2 ) and Q = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, aP + bP ′ + cQ is a rational divisor on the  curve if and only if a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  10  ALAIN COUVREUR  The group of divisors is equipped with a partial order relation denoted ⩽ and defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Given  two divisors  G =  �  P ∈X (K)  nP P  and  G′ =  �  P ∈X (K)  n′  P P,  we say that G ⩽ G′ if  ∀P ∈ X (K), nP ⩽ n′  P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In particular, a divisor G is said to be positive if G ⩾ 0, where 0 denotes the zero divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given a divisor G as in (4), its degree is defined as  deg G  def  = n1 + · · · + nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Given a function f ∈ K(X ) \\ {0}, one can associate its divisor denoted (f) and defined as  (5)  (f)  def  =  �  P ∈K(X )  vP (f)P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Such a divisor is called a principal divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For such an object to be a divisor, we need to show that the sum (5) is finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' that  the nP ’s are all zero but a finite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This is actually due to a well–known fact appearing in  the next statement whose proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A nonzero rational function on a curve has only a finite number of zeroes and  poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — It is worth noting that a principal divisor is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, one can first note that,  since f ∈ K(X ) and hence has its coefficients in K, then for any geometric point P ∈ X (K) and any  σ ∈ Gal(K/K) then vP (f) = vσ(P )(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The following very classical statement is crucial in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The degree of principal divisor is always 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We finish this discussion with a statement that we admit and which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A principal divisor (f) associated to f ∈ K(X )× is zero if and only if f is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Genus and Riemann–Roch Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The most elementary curve one may define is the  affine line A1 and its projective closure being the projective line P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Regular functions on A1 are nothing  but univariate polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Regarding such a polynomial h(x) ∈ K[x] as rational function on P1, it has  a pole at the point “at infinity”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' the points with homogeneous coordinates (1 : 0) and one can prove  that the valuation at this pole is nothing but − deg h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, the space K[x]⩽n of polynomials of degree less than or equal to n can be (with enough  pedantry) defined as the space of rational functions on P1 which are regular everywhere on an affine  chart and with valuation larger than or equal to −n at the point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Denoting by P∞ this point  at infinity, then the space K[x]⩽n can be regarded as the space of rational functions h ∈ K(P1) which are  either 0 or such that  (h) ⩾ −nP∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  As the following definition suggests, Riemann–Roch spaces are generalisations for curves of the spaces  K[x]⩽n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Definition 22 (Riemann–Roch space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X be a smooth projective absolutely irreducible  curve over K and G be a rational divisor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then the Riemann–Roch space associated to G is defined  as  L(G)  def  = {h ∈ K(X ) | (h) + G ⩾ 0} ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This is a vector space over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — According to the previous discussion, on P1, we have L(nP∞) ≃ K[x]⩽n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  11  The following statement summarises some properties of Riemann–Roch spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' —  (i) A Riemann–Roch space is a vector space over K of finite dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (ii) For any rational divisor G < 0, we have L(G) = {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (iii) For any rational divisor G, we have dimK L(G) ⩽ deg G + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  With the above statement at hand, we can introduce a fundamental invariant of a curve: its genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  There are dozens of manners to define this object but none of them is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The one given in these  notes is far from being satisfying since it is clearly not intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, it permits to define the object  with a minimal amount of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Definition 25 (Genus of a curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X be a smooth projective absolutely irreducible curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The genus of X is defined as  g = 1 − min  D {dimK L(D) − deg D},  where D ranges over all the divisors of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Proposition 24 (iii) asserts that the involved minimum exists and that the genus is  nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove the statement of Remark 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Using Definition 25, prove that the genus of the projective line P1 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that the effective computation of the genus is not a simple task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, for smooth plane  curves of degree d, there is a closed formula (see [Ful89, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5]):  g = (d − 1)(d − 2)  2 This permits in particular to prove that the projective line and smooth conics have genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The notion of genus can actually be defined for singular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this context, two  distinct invariants respectively called arithmetic genus and geometric genus can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These two  invariants coincide when the curve is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We conclude this section with Riemann–Roch Theorem, which is a crucial statement in the theory of  algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This statement is admitted and we refer the reader to Fulton [Ful89] or Stichtenoth’s  [Sti09] book for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The first part of the statement is actually a straightforward consequence of the  definition we gave for the genus (Definition 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 30 (Riemann–Roch Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X be a smooth absolutely irreducible curve of  genus g over K and G be a rational divisor on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then  dimK L(G) ⩾ deg G + 1 − g  and equality holds when deg G > 2g − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The Riemann–Hurwitz formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The last statement that will be useful in the sequel is  Riemann–Hurwitz formula which relates the genera of two smooth projective absolutely irreducible curves  X , Y linked by a non constant rational map ϕ : X ��� Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Recall that, according to Proposition 14,  such a map is regular and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Denoting by δ its degree (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4 for the definition of degree),  consider any geometric point P ∈ Y (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then one can prove that ϕ−1({P}) is a finite subset of X (K)  and that for any P but finitely many of them the cardinality of φ−1({P}) always equals δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The finite number of points of Y (K) where this no longer holds are called ramified points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Given  Q ∈ X (K), P = ϕ(Q) and t a local parameter at P, the ramification index of Q is defined as  eQ  def  = vQ(t ◦ ϕ),  t ◦ ϕ being an element of K(X ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It can be proved that this definition does not depend on the choice of  the local parameter t at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' According to the previous definition, for any point Q ∈ X (K) but finitely  many of them, we have eQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  12  ALAIN COUVREUR  Here we have the material to state Riemann–Hurwitz formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 31 (Riemann–Hurwitz formula (Tame version)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X , Y be two smooth projective  absolutely irreducible curves over K and ϕ : X ��� Y be a rational map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Suppose that for any Q ∈ Y (K),  the ramification index eQ is prime to the characteristic of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the genera gX , gY of X , Y are related  by the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (2gX − 2) = deg ϕ · (2gY − 2) +  �  Q∈Y (K)  (eQ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — According to the previous discussion, the terms of the sum in the above formula are  all zero but a finite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The assumption “ramification indexes are prime to the characteristic” can be discarded  at the cost of replacing the term �(eQ − 1) by a more complicated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' See [Sti09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This formula is particularly useful since many curves X are described by a morphism X → P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since  P1 is known to have genus 0, the genus of X can be deduced from the knowledge of the degree of this  map and the ramification indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Consider the map (2) of Example 13 but here we regard the curve C as a curve over  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' One sees that any point P = (t : 1) ∈ P1(C) has 2 preimages by the map if t /∈ {−1, 1} and only one if  t ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, there are two ramified points both with ramification index 2 (one can show that  the map does not ramify at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, the map has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, Riemann–Hurwitz formula  yields  2gC − 2 = 2(2gP1 − 2) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Since gP1 = 0, we deduce that gC = 0 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' What about non plane curves?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A last important fact is that some curves are not plane and  may be contained in PN for N > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It is actually important in the sequel since we are searching for  smooth curves X over a finite field Fq with ♯X (Fq) arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since ♯P2(Fq) is finite (and equal  to q2 +q +1) such a curve may not be embeddable in P2 and requires a larger dimensional ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  So, the question is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' what remains true when considering curves in PN with N > 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' and actually, how  are such objects defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We define a projective subvariety of PN as the common vanishing locus of the elements of a homoge-  neous ideal I ⊆ K[X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , XN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If this ideal is prime, then the variety will be said to be irreducible and  in this setting, the function field of the variety can be defined in the very same manner as in the plane  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the dimension of the variety can be defined as the transcendence degree of the function field  over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A curve will be a variety of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Smoothness can be defined very similarly by requiring  a non simultaneous vanishing of all the partial derivatives with respect to the N + 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' All the  other objects, rational maps, valuations, divisors, Riemann–Roch spaces can be defined in the very same  manner at the cost of heavier notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally all the previous statements on plane curves actually hold  for any curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometry codes  Now, we have the necessary material to define algebraic geometry (AG) codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Before, let us recall  the definition of Reed–Solomon codes that AG codes generalise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Reed–Solomon codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' —  Definition 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , αn be distinct elements of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let 0 ⩽ k ⩽ n, the code RSk is defined as  RSk(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , αn)  def  = {(p(α1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , p(αn)) | p ∈ Fq[x]⩽k−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  13  It is well–known that these codes have parameters [n, k, n − k + 1]q and hence reach the Singleton  bound (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, they are constrained in the sense that the αi’s should be distinct and hence the  length should be bounded by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, even if these codes have optimal parameters, it is hopeless to use  them in order to construct an infinite family of codes over a fixed field Fq whose length goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Here, curves enter the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Note first that Reed–Solomon codes may be defined in a much more pedant  manner as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider the projective line P1 and let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn be the rational points of P1 with  respective homogeneous coordinates (α1 : 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , (αn : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, RSk(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , αn) may be defined as  RSk(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , αn) = {(h(P1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , h(Pn)) | h ∈ L((k − 1)P∞)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This leads to a natural generalisation to algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The interest being the fact that a curve may  have more rational points than the projective line and hence replacing P1 by an arbitrary curve may  provide the opportunity of getting codes of length larger than q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometry codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — We give a minimal introduction to algebraic geometry (AG)  codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The reader interested in further references is encouraged to have a look at the surveys [HvLP98,  Duu08, CR21] or the books [TVN07, Sti09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We also refer to [HP95, BH08] for references on the  decoding of AG codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Definition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X be a smooth absolutely irreducible curve over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn) be  an ordered sequence of distinct rational points of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let G be a rational divisor on X whose support  avoids the points P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the algebraic geometry code CL(X , P, G) is defined as  CL(X , P, G)  def  = {(f(P1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , f(Pn)) | f ∈ L(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Once the codes are defined, their parameters can be evaluated using the previously introduced material  of algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let X be a smooth absolutely irreducible curve of genus g over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' let P = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn)  be a tuple of rational points of X and G be a rational divisor on X whose support avoids P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Suppose that deg G < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the parameters [n, k, d]q of CL(X , P, G) satisfy  k  ⩾  deg G + 1 − g  with equality when deg G > 2g − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (6)  d  ⩾  n − deg G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Denote by DP the divisor DP  def  = P1 + · · · + Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider the map  evP :  � L(G)  −→  Fn  q  f  �−→  (f(P1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , f(Pn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Its image is trivially CL(X , P, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The kernel of this map is the subspace of L(G) of functions f vanishing  at P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This subspace is nothing but L(G − DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' By assumption, deg(G − DP) = −(n − deg G),  is negative and hence, from Proposition 24 (ii), L(G − DP) = ker evP = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, evP is injective and  dim CL(X , P, G) = dim L(G) ⩾ deg G + 1 − g,  with equality if deg G > 2g − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Here, the last inequality together with the equality case are due to  Riemann–Roch Theorem (Theorem 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  For the minimum distance, let us introduce h ∈ L(G)\\{0} such that evP(h) has Hamming weight d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It  means that there exist distinct points Pi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pin−d among P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pn at which h vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consequently,  h ∈ L(G − Pi1 − · · · − Pin−d) and since h ̸= 0, the space L(G − Pi1 − · · · − Pin−d) ̸= {0}, which, from  Proposition 24 (ii) again, implies that deg(G − Pi1 − · · · − Pin−d) ⩾ 0 and hence  d ⩾ n − deg G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let us comment this last result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It was mentioned in § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2 that, from Singleton bound (1), any [n, k, d]q  code satisfies  k + d ⩽ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  14  ALAIN COUVREUR  On the other hand, Theorem 37 asserts that an [n, k, d]q AG code CL(X , P, G) satisfies  n + 1 − g ⩽ k + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In summary, AG codes are in the worst case at “distance g from Singleton bound”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, one can expect  good codes for a “not too large” genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On the other hand, the objective is to construct sequences of  codes whose length exceeds q and more generally construct families of codes over Fq whose length goes to  infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, for the length to be large, we look for curves with the largest possible number of rational  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The problem of infinite sequence of curves with many points compared to their genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  — We expect to get sequences of curves over Fq whose genus grows slowly and number of rational points  grows quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, these two objectives are somehow in opposition: to get many rational points,  we need a large genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A well–known result due to Weil asserts that for a smooth absolutely irreducible  curve X over Fq,  (8)  ♯X (Fq) ⩽ q + 1 + 2g√q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Thus, we look for a good trade off between the genus and the number of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Now, we have the  material to reformulate our coding theoretic problem of producing asymptotically good infinite sequences  of codes in terms of the construction of sequences of algebraic curves with specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For this, let  us consider a sequence of curves (Xs)s∈N with sequence of genera (gs)s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We suppose that the sequence  (♯Xs(Fq))s∈N goes to infinity, hence, according to Weil’s bound (8), the sequence of genera should also  go to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let  (9)  γ = lim sup  s→+∞  ♯Xs(Fq)  gs For any such curve in the sequence, we fix a rational divisor Gs and the sequence of rational points  Ps = (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pns) will be chosen as the full list of rational points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' ns = ♯Xs(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — One could ask whether it is possible to have a rational divisor Gs of any degree whose  support avoids P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pns while {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , Pns} = X (Fq)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The answer is positive, such divisors Gs exist  and the constraint that the support of Gs should avoid X (Fq) is actually easy to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' See [CR21,  Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8] for a detailed discussion on this specific question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Then, the codes CL(Xs, Ps, Gs) have parameters [ns, ks, ds]q satisfying  ns  =  ♯Xs(Fq)  ks  ⩾  deg Gs + 1 − gs  ds  ⩾  ns − deg Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, one can eliminate deg Gs and get  (10)  ks + ds ⩾ ns + 1 − gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Set  R = lim sup  s→+∞  ks  ns  and  δ = lim sup  s→+∞  ds  ns Then, dividing (10) by ns and letting s go to infinity, we get  R + δ ⩾ 1 − 1  γ ,  where γ is defined in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, any pair (δ, R) lying on the line of equation R + δ = 1 − 1  γ is  achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Even if the term deg Gs has been eliminated, this term is worth in order to chose the  point in the line of equation R + δ = 1 − 1  γ you want to target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove that by choosing a relevant sequence of rational divisors (Gs) on the curves Xs,  one can reach any point of the line of equation R + δ = 1 − 1  γ ·  CODES AND MODULAR CURVES  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The Ihara constant A(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Now, we would like to estimate the optimal asymptotic parameters  (δ, R) that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For that, let us introduce the Ihara constant:  A(q)  def  = lim sup  g→+∞  max  X , curve  of genus g  ♯X (Fq)  g Then, the Tsfasman–Vlăduţ–Zink (TVZ) bound asserts the existence of families of codes whose asymp-  totic parameters (δ, R) satisfy  R + δ ⩾ 1 −  1  A(q)·  This opens the question of the value of A(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The remainder of these notes consists in outlining a proof  of the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For q = p2 and p a prime number, we have  A(q) ⩾ √q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Combining the previous result with the TVZ bound, one ca prove that for p ⩾ 7, and hence when q is  the square of a prime and is larger than or equal to 49, the TVZ bound exceeds the Gilbert Varshamov  one, proving that some families of codes from algebraic curves are better than random codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let us conclude with some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Actually, the result extends to q = p2m for any m ⩾ 1 but the proof gets more complicated and  involves other families of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely, the proof to follow involves modular curves, while the  general case involves Shimura curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' See [Iha81, TVZ82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The TVZ bound is actually optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, subsequently to the publication of Tsfasman–Vlăduţ–  Zink result, in [VD83] Drinfeld and Vlăduţ proved that for any prime power q, we always have  A(q) ⩽ √q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Another proof of Theorem 41 using a very different approach has been given by Garcia and  Stichtenoth in [GS95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The core of the proof of this wonderful result rests on the use of families of curves called modular  curves which parameterise families of algebraic curves called elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Elliptic curves  Elliptic curves is another fascinating topic in number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' They are also a fundamental object  in cryptography but this is not the point of these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this section, we start by presenting basic  notions about these objects over an arbitrary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Our objective is in particular to construct these  so–called modular curves which will yield excellent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These modular curves are algebraic curves  which parameterise families of elliptic curves with a specific extra structure called level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Afterwards, in Section 5, we will discuss elliptic curves and modular curves over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This choice of  discussing complex curves in such notes might seem surprising while our interest will clearly be curves  over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, a preliminary study of the complex case presents several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' First, it  provides a much more intuitive presentation of the topics with the benefits of the possible use of analytical  tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Second, even if the analytic proofs cannot transpose in the finite field setting, they permit to  compute algebraic formulas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' polynomial equations defining modular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These equations turn  out to be defined over Z and then — and this is very far from being trivial — their reduction modulo p  will give the equation of a curve parameterising elliptic curves over Fp or Fp with some level structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this section, we assume the ground field K to have characteristic different from 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Most  of the material of the present section and the subsequent one are taken from the book [Sil09] and the  lecture notes [Mil17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — An elliptic curve E over a field K is a smooth projective curve of genus 1  with at least one rational point denoted by OE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Proposition 21, the Riemann–Roch space L(0)  associated to the zero divisor contains only the constant functions and hence has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, by  Riemann–Roch Theorem, the spaces L(2OE ) and L(3OE ) have respective dimensions 2 and 3 (note that  16  ALAIN COUVREUR  as soon as the divisor’s degrees are positive, they are > 2g − 2 and hence we fit in the equality case of  Riemann–Roch Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Denote by x, y two functions such that  L(2OE ) = SpanK{1, x}  L(3OE ) = SpanK{1, x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that these choices for x and y are not canonical and hence any of the following changes of variables  are admissible  (11)  x′ ← ax + b, with a ̸= 0  y′ ← uy + vx + w, with u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Now, consider the space L(6OE ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It contains the functions  1, x, y, x2, xy, x3, y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Moreover, again from Riemann–Roch Theorem, L(6OE ) has dimension 6 and hence there is a nontrivial  linear relation on these functions  y2 + uxy + vy = ax3 + bx2 + cx + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove that y2 and x3 should be involved in this linear relation, which explains why,  after a renormalisation, one can suppose the coefficient of y2 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Deduce from this that a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Now, we perform successive changes of variables which are admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' changes of variables of the  form (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A first one(7): y ← y + u  2 x leads to an equation:  (12)  y2 + v1y = a1x3 + b1x2 + c1x + d1,  for some a1, b1, c1, d1 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A change y ← y + v1  2 yields  (13)  y2 = a2x3 + b2x2 + c2x + d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  for some a2, b2, c2, d2 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, a change of the form x ← x +  b2  3a2 yields to an equation:  (14)  y2 = a3x3 + c3x + d3,  for some a3, c3, d3 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally, applying the change of variables x ← a3x, y ← a2  3y and dividing both  sides by a4  3, we get an equation of the form  (15)  y2 = x3 + Ax + B,  for some A, B ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such an equation is called a Weierstrass equation of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Using Exercise 42, check that the last change of variables was admissible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' that  a3 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In summary, starting from an elliptic curve E over K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' a smooth genus 1 curve with a rational  point OE , we found two functions x, y ∈ K(E ) which are both regular everywhere but at OE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' These  functions are related by the relation (15) and hence the function y2 − x3 − Ax − B vanishes everywhere  on E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This leads to the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let E be an elliptic over a field K of characteristic different from 2 and 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' a  smooth projective curve of genus 1 with a rational point OE , then there exist x, y ∈ K(E ) such that the  map  �  �  �  E  ���  P2  P  �−→  � (x(P) : y(P) : 1)  if  P ̸= OE  (0 : 1 : 0)  if  P = OE  induces an isomorphism from E to the projective closure of the projective curve of equation Y 2 = X3 +  AXZ2 + BZ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (7)In order to keep light notation, we remove the ’ in x′, y′ and hence write the outputs of the change of variables as the  input, hence the notation y ← y + u  2 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This is not a completely rigorous notation and the reader bothered by this is  encouraged to rewrite this page by replacing the x’s and y’s as x′, x′′, x′′′ and y′, y′′, y′′′ at the good spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The fact that the image of E is contained in such a curve is a consequence of the previous  discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' To prove that this map is actually an isomorphism and in particular that the target curve is  smooth, we refer the reader to [Sil09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Geometrically speaking, the sequence of changes of variables can be interpreted as  follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We started from an elliptic curve E and a first choice of functions x, y in K(E ) lead to an  isomorphism between E and a curve with equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, we applied successive affine automorphisms  to the plane in order to get curves of successive equations (13), (14) which are pairwise isomorphic and  finish with a curve with equation (15) which is also isomorphic to E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the sequel, we will not only consider Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Actually, one can show that  any curve of equation  y2 = f(x)  where f is a squarefree polynomial of degree 3 is an elliptic curve and there is a change of variables  permitting to put it in Weierstrass form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The j–invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In what follows, it will be important to classify elliptic curves up to isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For this sake, we introduce a fundamental invariant: the j–invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Reconsider a Weierstrass  equation (15)  y2 = x3 + Ax + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This equation is not unique, since once we got it, one can still apply changes of variables of the form  y ← u3y, x ← u2x and dividing both sides by u6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This leads to another Weierstrass equation y2 =  x3 + A′x + B′ where A′ = A  u4 and B′ = B  u6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let us introduce  j  def  = 1728  4A3  4A3 + 27B2 ·  This quantity is well–defined since one can prove that the denominator 4A3 + 27B2 is zero if and only  if the corresponding curve is singular (see [Sil09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4(a)(i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence, j is well–defined for any  elliptic curve since, by definition, such curves are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, j is left invariant by the previous  change of variables and one can show that, once we obtained a Weierstrass equation, the only changes of  variables preserving the Weierstrass equation structure are the aforementioned ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We conclude this subsection by the following statements asserting that the j–invariant characterises an  elliptic curve over K in a unique manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The proof is omitted and can be found in [Sil09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4(b-  c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Two elliptic curves are isomorphic over K if and only if they have the same j–  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Conversely, given j0 ∈ K, there exists an elliptic curve E over K(j0) with j–invariant j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Note that two elliptic curves defined over K may be isomorphic over K without being  isomorphic over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For instance, suppose that −1 is not a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, between the curves with  equation  y2 = x3 + Ax + B  and  − y2 = x3 + Ax + B  are related by the isomorphism defined over K given by (x, y) �→ (x, √−1 y) but there may not exist an  isomorphism defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such curves are said to be a twist of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Starting from a j–invariant j0 ∈ K, an explicit equation for an elliptic curve with this  j–invariant is given by  y2 + xy = x3 −  36  j0 − 1728x −  1  j0 − 1728  if  j0 ̸= 0, 1728  and  y2 + y = x3 if j0 = 0  and  y2 = x3 + x if j = 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  18  ALAIN COUVREUR  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The addition law on an elliptic curve (Source: Cornelius Schätz blog)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The group law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A remarkable feature of such curves is that they naturally have a group  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely, given an elliptic curve E over K, the set E (K) has a structure of abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' More  generally, for any algebraic extension L of K, then E (L) has an abelian group structure too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This group  structure is usually represented with a so–called chord and tangent process as represented by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  It can be described as follows:  Given two points P, Q ∈ E (L), draw the line L ⊆ A2 (or P2) joining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If P = Q let L be the  tangent line of E at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Since the curve has degree 3, its intersection with L and E has 3 points counted with multiplicity  and hence either L is vertical and then the third point is R0 = OE or denote by R0 = (xR0, yR0)  be the third(8) point of intersection of this line with E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let R be the point with coordinates (xR0, −yR0) if R0 ̸= OE and the point OE otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This  point is defined to be the sum of P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — (1) Prove that the intersection of E with a line is made of 3 points of P2 possibly  counted with multiplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (2) Prove that R0 ∈ E (L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This description is simple to understand but it is not completely obvious to prove that it provides a  group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In particular, the associativity is far from being obvious using this description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Here we  will show that this group structure can also be understood as a group law inherited from that of some  quotient of the divisor group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, denote by DivK(E ) the group of rational divisors on E and by  Div0  K(E ) the subgroup of divisors of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally denote by PrincK(E ) the group of principal divisors,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' of divisors of the form (f) where f ∈ K(E ) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Remark 19 together with Proposition 20,  PrincK(E ) is a subgroup of Div0  K(E ) and the quotient is denoted  Pic0  K(E )  def  = Div0  K(E )/PrincK(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let E be an elliptic curve over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Any element of Pic0  K(E ) has a representative of  the form P − OE where P is some rational point in E (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let G ∈ Div0  K(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The divisor G + OE has degree 1 and, from Riemann–Roch Theorem  L(G + OE ) has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, there exists f ∈ L(G + OE ) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' By definition of L(G + OE ), the  function f satisfies  (f) + G + OE ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The latter divisor is positive with degree 1 and hence equals some rational point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus (f)+G = P−OE ,  which entails that G and P − OE have the same class in Pic0  K(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (8)Possibly R0 equals P or Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This is the reason why we mentioned 3 points counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If the intersection  multiplicity of L with E at P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Q) is 2 then, we set R0  def  = P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  R  R=P+QCODES AND MODULAR CURVES  19  Theorem 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let P, Q ∈ E (K) and R be the sum of P + Q according to the previously introduced  addition law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the classes of R − OE and (P − OE ) + (Q − OE ) are the same in Pic0  K(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — From Exercise 50 (1), there is a point R0 which is contained in the line L joining P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Moreover R, R0 are contained in a vertical line L ′, the verticality entails that, projectively speaking, R0, R  and OE are in the projective closure of the line L ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let H(X, Y, Z) and H′(X, Y, Z) be homogeneous  polynomials of degree 1 providing equations of the projective closures of L and L ′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The  rational function h  def  =  H  H′ ∈ K(E ) has divisor  (h) = (P + Q + R0) − (R0 + R + OE ) = (P − OE ) + (Q − OE ) − (R − OE ),  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  As a conclusion, we have the following bijection:  �  E (K)  −→  Pic0  K(E )  P  �−→  P − OE  mod PrincK(E ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Via this bijection, we can equip E (K) with a group structure whose law is nothing but the previously  described chord–tangent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, E (K) equipped with the chord–tangent law has a group structure  which is isomorphic to Pic0  K(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Here again, note that we discussed about the group structure of the set of rational  points E (K) but actually, for any algebraic extension L/K, the set E (L) has also a group structure with  E (L) as a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In particular, the whole set of geometric points E (K) has a structure of abelian  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Torsion and isogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Once we know that elliptic curves are equipped with an abelian group  structure it is of course natural to study the morphisms relating these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For this sake, we first need  to discuss some specific subgroups of points of elliptic curves: their torsion subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Torsion subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Given an elliptic curve E and an integer ℓ, one is interested in the group  E [ℓ]  def  = {P ∈ E (K) | ℓP = 0},  where ℓP means “P + · · · + P” (added ℓ times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Interestingly, this group has a natural structure of  Z/ℓZ–module, and, in particular, is an Fℓ–vector space when ℓ is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The next theorem asserts that  this space has always dimension 2 when ℓ is prime to the characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The proof of the next statement  is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let E be an elliptic curve over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let ℓ be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If ℓ is prime to the characteristic  of K, then  E [ℓ] ≃ Z/ℓZ × Z/ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Else, if p denotes the characteristic of K and p ̸= 0, then  E [p] ≃  � either  Z/pZ  or  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In the former case the curve is said to be ordinary, in the latter it is said to be supersingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Isogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Given two elliptic curves E , E ′, an isogeny φ : E → E ′ is a morphism between these  curves sending the neutral element OE onto OE ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such a map is always surjective from E (K) into E ′(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A remarkable property is that such a map is necessarily a morphism of groups (see [Sil09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  As any morphism of curves, an isogeny φ : E → E ′ induces a field extension K(E ′)/K(E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The degree of  the isogeny is the degree of the field extension and the isogeny is said to be separable if the field extension  is separable too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An isogeny of degree ℓ will usually be referred to as an ℓ–isogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Taken from [Sil09, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let a, b ∈ K, b ̸= 0 and a2 − 4b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider the  curves with equations:  E : y2  =  x3 + ax2 + bx  E ′ : y2  =  x3 − 2ax2 + (a2 − 4b)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  20  ALAIN COUVREUR  The following map is a 2–isogeny:  (16)  �  E  −→  E ′  (x, y)  �−→  �  y2  x2 , y(b−x2)  x2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Check that the map (16) actually sends E into E ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Using a computer algebra  software may be helpful for this exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Another example for isogenies of elliptic curves over a finite field Fq of characteristic  p is the Frobenius map  �  E  −→  E (p)  (x, y)  �−→  (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This isogeny is purely inseparable and sends the curve E with Weierstrass equation y2 = x3 + Ax + B  onto the curve E (p) of equation y2 = x3 + Apx + Bp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If E is defined over Fq (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' if A, B ∈ Fq) then the  Frobenius map is an endomorphism of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For any m > 0 prime to the characteristic and any elliptic curve E over K, the map  P �→ mP is an isogeny from E into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Its kernel is E [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A separable isogeny of degree ℓ, regarded as a group morphism E (K) → E (K) is surjective with a  finite kernel of cardinality ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Its kernel is a subgroup of E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 59 ([Sil09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For any finite subgroup K ⊆ E (K), there exists an elliptic  curve E ′ defined over K and an isogeny φ : E → E ′ such that ker φ = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The curve E ′ is sometimes  denoted as E /K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the previous statement, further precision can be given on the field of definition of  E ′ and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The field of definition of the group K is the smallest extension L/K such that K is globally  invariant under the action of Gal(K/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The field of definition of φ and E ′ is that of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that the field of definition is not the smallest field of definition of any geometric point of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For  instance, there may be non rational m–torsion points while E [m] is defined over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, even if an isogeny φ : E → E ′ of degree m > 1 is not an isomorphism in general, and hence  has no inverse, it has a so–called dual isogeny ˆφ which is the unique isogeny ˆφ : E ′ → E such that  φ ◦ ˆφ :  � E  −→  E  P  �−→  mP  and  ˆφ ◦ φ :  � E ′  −→  E ′  Q  �−→  mQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The existence and uniqueness of this map are proven in [Sil09, § III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the case of a separable isogeny φ : E → E ′ of degree m, its kernel is a group with  m elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' By Lagrange Theorem, such a finite group is of m–torsion and hence ker φ ⊆ E [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then  φ(E [m]) is a finite subgroup and, from Theorem 59, there is an isogeny ϕ : E ′ → E ′/φ(E [m]), which is  nothing but the dual isogeny of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In particular E ′/φ(E [m]) ≃ E /E [m] ≃ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The last isomorphism is  induced by the map P �→ mP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  All the previously introduced notions: torsion, isogenies, dual isogenies will be re–discussed and better  illustrated in the subsequent section about elliptic curves over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this context, these notions will be  much easier to visualise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Elliptic curves over the complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — As already explained earlier, complex elliptic  curves is not the topic of this lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It is however necessary to discuss a bit about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In order not  to spend too much time on the topic, many proofs of non trivial statements are omitted and replaced by  precise references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Clearly, the summary to follow is strictly included in Chapter VI of Silverman’s book  [Sil09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Lattices and the Weierstrass ℘ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the complex setting, an elliptic curve is isomorphic  to a complex torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely, a lattice of C is a discrete subgroup Λ with compact quotient and it is well–  known that such a group is of the form  Λ = Zω1 ⊕ Zω2  where ω1, ω2 are linearly independent over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The relation between a torus C/Λ and an elliptic curve is  far from being obvious and the key for connecting these two objects is Weierstrass ℘Λ function defined  as  ℘Λ(z)  def  =  1  z2 +  �  ω∈Λ\\{0}  �  1  (z − ω)2 − 1  ω2  � This is a meromorphic function with pole locus Λ which is Λ–periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' for any z ∈ C \\ Λ and ω ∈ Λ,  ℘(z + ω) = ℘(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The proof of convergence of the series is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that, since ℘ is Λ–periodic, it passes to the quotient and induces a meromorphic function on  the torus C/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The function ℘ is fundamental in the sense that actually, any Λ–periodic meromorphic  function can be expressed as a rational function in ℘ and its derivative ℘′ as explained by the following  statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — There exist complex numbers g2, g3, which depend on Λ such that  ∀z ∈ C \\ Λ,  ℘′  Λ(z)2 = 4℘Λ(z)3 + g2℘Λ(z) + g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The series  ℘(z) − 1  z2 =  �  ω∈Λ\\{0}  �  1  (z − ω)2 − 1  ω2  �  is even and vanishes at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence, in the neighbourhood of 0, its Taylor series expansion depends only on  z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, we deduce that ℘Λ has a Laurent series expansion at 0 of the form  ℘Λ(z) = 1  z2 + O(z2),  and  ℘′  Λ(z) = − 2  z3 + O(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, in a neighbourhood of 0, ℘′  Λ(z)2 − 4℘Λ(z)3 = O( 1  z2 ) and there is a constant g2 ∈ C such that  (17)  ℘′  Λ(z)2 − 4℘Λ(z)3 − g2℘Λ(z) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The function ℘′  Λ(z)2 − 4℘Λ(z)3 − g2℘Λ(z) is Λ–periodic, meromorphic on C with pole locus contained  in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From (17), it has no pole at 0 and, by Λ–periodicity has no pole at all and hence is holomorphic  on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since it continuous and Λ–periodic on C, it is bounded, and by Liouville’s theorem, it should be  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, there exists g3 ∈ C such that ℘′  Λ(z)2 = 4℘Λ(z)3 + g2℘Λ(z) + g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove that a Λ–periodic holomorphic function is bounded on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  A finer analysis of the series permits to estimate g2, g3 in terms of Λ and to prove that the equation  y2 = 4x3 + g2x + g3 is that of a smooth curve, and hence of an elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  With this theorem  at hand, we deduce the existence of a map from the torus C/Λ into the elliptic curve E of equation  y2 = 4x3 + g2x + g3:  (18)  ΨΛ :  � C/Λ  −→  E  z  �−→  (℘Λ(z) : ℘′  Λ(z) : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that this map is well–defined everywhere, since at 0 which is a pole of order 2 of ℘Λ and of order  3 of ℘′  Λ one can renormalise as (z3℘Λ(z) : z3℘′  Λ(z) : z3) and evaluate at 0, which yields the point  OE = (0 : 1 : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The following statement gathers several nontrivial and fundamental results on complex  tori: it states a one-to-one correspondence between elliptic curves and complex tori when regarded as  complex varieties but also as groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The map ΨΛ defined in (18) is a biholomorphic isomorphism between C/Λ and the  elliptic curve E of equation y2 = 4x3+g2x+g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, it also induces a group isomorphism from C/Λ  22  ALAIN COUVREUR  equipped with the addition law inherited from that of C into E (C) equipped with its group law introduced  in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Conversely, given any elliptic curve E0 over C, there exists a lattice Λ0 ⊂ C such that E0 is  isomorphic to C/Λ0 via the map ΨΛ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — See [Sil09, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6] for the group isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For the construction of a lattice from an  elliptic curve, see [Sil09, § VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Torsion, isogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — An interest of the complex setting is that the previous results on torsion  and isogenies are pretty easy to understand when regarding elliptic curves as complex tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let us start with the torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Theorem 54, for m prime to the characteristic, the m–torsion of  an elliptic curve is isomorphic to Z/mZ × Z/mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the complex setting, consider a torus C/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then,  the torsion points correspond to points z ∈ C such that mz ∈ Λ and hence they correspond to the points  of the lattice  1  mΛ ⊃ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, the torsion subgroup of C/Λ is isomorphic to  1  mΛ/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since Λ is of the  form Zω1 ⊕ Zω2 for some R–independent elements ω1, ω2 ∈ C, we deduce that  (C/Λ)[m] ≃  � 1  mΛ  �  /Λ = Z ω1  m ⊕ Z ω2  m  Zω1 ⊕ Zω2  ≃ Z/mZ ⊕ Z/mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Now consider isogenies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' When considering complex tori, isogenies are holomorphic maps C/Λ → C/Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  It turns out that such maps lift to C and have a very particular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let Λ, Λ′ ⊂ C be two lattices and f : C/Λ → C/Λ′ be a holomorphic map 0 sending  onto 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then f lifts to a holomorphic map f0 : C → C such that  ∀z ∈ C,  f0(z)  mod Λ′ = f(z  mod Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Moreover, f0 is a similitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' there exists a ∈ C such that  ∀z ∈ C, f0(z) = az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — See [Sil09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  With this statement at hand, we deduce that an isogeny φ : C/Λ → C/Λ′ is induced by a map z �→ az  with aΛ ⊆ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Example 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For instance, consider the lattices  Λ = Z ⊕ Z2i  and  Λ′ = 2Z ⊕ Z2i  then we easily see that the map z �→ 2z induces an isogeny C/Λ → C/Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  From a similitude z �→ az such that aΛ ⊂ Λ′, the degree of the corresponding isogeny is given by  ♯(Λ′/aΛ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the previous example, the isogeny has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, let ℓ be a prime integer, and suppose that we have a degree–ℓ isogeny φ1 : C/Λ → C/Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This  entails the existence of a ∈ C such that ♯(Λ′/aΛ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From the structure theorem of finitely generated  modules over principal ideal rings, we deduce the existence of ω1, ω2 ∈ C such that  Λ = Zℓω1  a  ⊕ ω2  a ,  Λ′ = Zω1 ⊕ Zω2  and φ1 is induced from the similitude z �→ az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove the last assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Now consider the map z �→ ℓ  az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It induces an isogeny φ2 : C/Λ′ → C/Λ of degree ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover the  composition of the two isogenies :  φ2 ◦ φ1 : C/Λ → C/Λ′ → C/Λ′′  is defined by z mod Λ �→ ℓz mod Λ and hence is nothing but the multiplication by ℓ in C/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore,  φ2 is nothing but the dual isogeny map ˆφ1 of φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Now, we have a nice description of morphisms of complex elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Moreover, an endomorphism of an elliptic curve, or equivalently of a complex torus C/Λ, is induced by  a similitude z �→ az such that aΛ ⊂ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  One will also be interested in the sequel by automorphisms of an elliptic curve, which correspond to  similitudes z �→ az such that aΛ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' One can prove that such an a satisfies |a| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Clearly for any integer N and any lattice Λ we have NΛ ⊂ Λ and the map z �→ Nz  induces an endomorphism of C/Λ which is the multiplication by N map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In addition, if there exists  a ∈ C\\Z such that aΛ ⊂ Λ, then the corresponding elliptic curve is said to be with complex multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  An elementary automorphism for any complex torus is z �→ −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Back to the map ΨΛ in (18) and  using the fact that ℘Λ and ℘′  Λ are respectively even and odd, we deduce that this map corresponds on  the elliptic curve to the symmetry with respect to the x–axis:  (x, y) �−→ (x, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Furthermore, some sporadic elliptic curves have nontrivial automophisms coming from z �→ az with  |a| = 1 and a /∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let C/Λ be a complex torus with an automorphism z �→ az with |a| = 1 and a /∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Equivalently, the lattice Λ satisfies Λ = aΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, Λ is the image by a similitude of one of these two  lattices:  Z ⊕ Zi  or  Z ⊕ Zρ,  where ρ = e  iπ  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let Λ be a lattice such that aΛ = Λ and ν ∈ Λ \\ {0} be a vector of minimal modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since we  look for Λ up to a similitude, one can assume that ν = 1 and that for all ω ∈ Λ \\ {0}, |ω| ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Assuming  that 1 ∈ Λ, then, by assumption on Λ, we deduce that a, a2 are elements of Λ too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since a /∈ R, its  minimal polynomial over R is  (x − a)(x − ¯a)  =  x2 + 2Re(a)x + |a|2  =  x2 + 2Re(a)x + 1,  where Re(a) denotes the real part of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consequently,  a2 + 1 = −2Re(a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that |a| = 1 and a /∈ R entails −1 < Re(a) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If 2Re(a) /∈ Z, then there is ε ∈ {−1, 0, 1} such that  a2 + εa + 1 = γa  for some 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since the left–hand side is a Z–linear combination of elements of Λ, then γa ∈ Λ  which contradicts the assumption that any nonzero ω ∈ Λ satisfies |ω| ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore Re(a) ∈ {− 1  2, 0, 1  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Case Re(a) = 0 provides the case Λ = Z ⊕ Zi and the two other cases provide the same lattice, namely  Z ⊕ Zρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The corresponding elliptic curves can be proved to have respective equations:  (19)  y2  =  x3 + x  for  Λ  =  Z ⊕ Zi  (j–invariant 1728)  y2  =  x3 + 1  for  Λ  =  Z ⊕ Zρ  (j–invariant 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The corresponding automorphisms being respectively  (x, y)  �−→  (−x, iy)  (x, y)  �−→  (ρx, −y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that these automorphisms have respective orders 4 and 6 which are the multiplicative orders of i and  ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally, note that for any field containing fourth and sixth roots of 1, the curves with equations (19)  have a nontrivial automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, one can prove that they are the only curves with non  trivial automorphism groups [Sil09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1] and that their automorphism groups have respective  cardinalities 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  24  ALAIN COUVREUR  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Modular curves  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The Poincaré upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The objective is to classify elliptic curves over C up to  isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' As explained in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5, this reduces to classify lattices up to similitudes whose definition is  recalled there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Definition 70 (Similitudes of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A similitude of C is a map of the form z �→ az for some a ∈ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Besides the action of the group of similitudes on the set of lattices of C, lattices are described by a  basis which is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This requires to introduce another group action on the possible bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely,  given a lattice  Λ = Zω1 ⊕ Zω2,  the basis (ω1, ω2) is not unique and any other basis (µ1, µ2) is deduced from (ω1, ω2) by  �µ1  µ2  �  = M ·  �ω1  ω2  �  ,  for some M ∈ GL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Up to swapping the entries of the basis, one can always assume that the bases we consider have the  same orientation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' that Im( ω1  ω2 ) > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Im( µ1  µ2 ) > 0), where Im(·) denotes the imaginary part of  a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If the bases are chosen under this constraint, then the transition matrix M always  has a positive determinant and hence is in SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the set of lattices of C is in one-to-one  correspondence with the classes of pairs (ω1, ω2) ∈ C2 with Im( ω1  ω2 ) > 0 modulo the action of SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Next, we need to consider the action of similitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Starting from Λ = Zω1 ⊕ Zω2 with Im( ω1  ω2 ) > 0 and  applying the similitude z �→  1  ω2 z, we get a similar lattice:  Z ⊕ Zτ  with τ = ω1  ω2 and hence Im(τ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let  H  def  = {z ∈ C | Im(z) > 0} ,  be the Poincaré upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then any lattice up to similitude can be associated to an element  τ ∈ H and the action of SL2(Z) on bases of lattices induces the following action on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Starting from  M =  �a  b  c  d  �  ∈ SL2(Z),  M acts on bases as:  M ·  �ω1  ω2  �  =  �aω1 + bω2  cω1 + dω2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore since τ = ω1  ω2 , we naturally define the action of SL2(Z) on H by  (20)  M · τ  def  = aω1 + bω2  cω1 + dω2  = aτ + b  cτ + d·  In summary, according to the discussion of § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5, we have the following correspondence:  Elliptic curves  Complex tori  Lattices of C  Points of H  up to  ←→  up to  ←→  up to  ←→  modulo  isomorphism  biholomorphic  similitudes  the action (20) of  isomorphisms  SL2(Z)  Moreover, fundamental domains for the action of SL2(Z) on H are represented in Figure 4, which is a  famous picture that you can find in so many books of geometry or number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The curve X0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — So, to parameterise the set of elliptic curves up to isomorphism, we can  consider the quotient SL2(Z)\\H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It is proved in [Mil17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='21] that this quotient is a complex  variety isomorphic to A1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' to the complex affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This is not surprising, Theorem 65 entails  that complex elliptic curves up to ismomorphisms are in one-to-one correspondence with C via the map  E �→ j(E ), where j(E ) denotes the j–invariant of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  25  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Fundamental domain for the action of SL2(Z) on H (Source: Wikipedia)  Next, for convenience and in order to apply results on algebraic curves introduced in § 2, it will be  useful to have some projective closure of this parameterising curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the complex setting, this is nothing  but a compactification and the affine line can be compactified with one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, for a reason  which will appear to be more natural in the sequel, the compactification will be made via a somehow  more complicated construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The idea is to join to H all the elements of Q which lie on the boundary of H together with a point at  infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Namely, we define  H∗ def  = H ∪ P1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Next let us see how the action of SL2(Z) extends to P1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Consider the following action of SL2(Z) on P1(Q):  ∀M =  �a  b  c  d  �  ∈ SL2(Z), (u : v) ∈ P1(Q),  M · (u : v) = (au + bv : cu + dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This action is transitive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' for any (u : v), (u′ : v′) ∈ P1(Q), there exists M ∈ SL2(Z) such that  M · (u : v) = (u′ : v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — First, let us prove that the orbit of (0 : 1) equals the whole P1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Note first that  �0  −1  1  0  �  (0 : 1) = (−1 : 0) = (1 : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Hence (1 : 0) is in the orbit of (0 : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, consider any other point (s : t) ∈ P1(Q) \\ {(1 : 0)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' such  that t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' After multiplying the coordinates by a common denominator, one can suppose that s, t ∈ Z  and after possibly dividing by their greatest common denominator, one can suppose s, t are prime to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' By Bézout’s Theorem, there exist u, v ∈ Z such that su + tv = 1 and then  � t  u  −s  v  �  (0 : 1) = (u : v)  and  � t  u  −s  v  �  ∈ SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, any element of P1(Q) is in the orbit of (0 : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, given two elements (u : v), (u′ :  v′) ∈ P1(Q) there exist M, M ′ such that (u : v) = M · (0 : 1) and (u′ : v′) = M ′ · (0 : 1) and  (u′ : v′) = M ′M −1(u : v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, the quotient SL2(Z)\\H∗ is nothing but the compactification of SL2(Z)\\H by adjoining a  single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This quotient is usually denoted as X0(1) and is nothing but the Riemann sphere P1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  In terms of functions on X0(1), there exists a holomorphic function j : H → C which is invariant under  the action of SL2(Z) and such that the induced map SL2(Z)\\H → C is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This map realises an  isomorphism between X0(1) and P1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It can be “made explicit” as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From τ ∈ H construct the  lattice Λτ = Z ⊕ Zτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then using the Weierstrass ℘Λτ function, compute an equation of the elliptic curve  corresponding to C/Λτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, j(τ) is nothing but the j–invariant of this latter elliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  2  1  0  1  226  ALAIN COUVREUR  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The curve X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Once we have a curve parameterising elliptic curves up to isomorphisms, we  are still a bit far from our objective since we look for a family of curves whose sequence of genera goes to  infinity, while we only got P1 which has genus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' To get curves with a higher genus, we need to enhance  the structure and the idea is not only to classify elliptic curves up to isomorphism but to classify for a  fixed integer ℓ, the ℓ–isogenies E → E ′ up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In the sequel we are only interested in the  case where ℓ is prime (but many of the results to follow extend to an arbitrary degree of isogeny).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Note that, here, by “up to ismomorphism” we mean that two isogenies φ1 : E1 → E ′  1 and  φ2 : E2 → E ′  2 will be said to be isomorphic if there exist two isomorphisms η : E1 → E2 and ν : E ′  1 → E ′  2  such that the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  E1  E ′  1  E2  E ′  2  η  ν  φ1  φ2  From Theorem 59, an ℓ–isogeny E → E ′ corresponds to a pair (E , C) where C ⊆ E [ℓ] is a subgroup of  cardinality ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, in the complex setting, it reduces to classify pairs of lattices Λ, Λ′ such that Λ ⊆ Λ′  and ♯(Λ′/Λ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The structure theorem for finitely generated modules over a principal ideal ring asserts  that there exists a basis ω1, ω2 of Λ such that  Λ = Zω1 ⊕ Zω2  and  Λ′ = Zω1  ℓ ⊕ Zω2·  With the above description, one sees easily that E [ℓ] = ( 1  ℓ Λ)/Λ ≃ Fℓ ⊕ Fℓ and Λ′/Λ identifies to an  Fℓ–subspace of dimension 1 of E [ℓ], namely the subspace spanned by the class of ω1  ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since we wish to  classify elliptic curves E with a given ℓ–torsion subgroup C, we need to classify changes of basis preserving  this subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Observe that the action of SL2(Z) on bases of Λ induces a natural action of SL2(Fℓ) on  E [ℓ] = ( 1  ℓ Λ)/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The elements of SL2(Fℓ) that fix the class of ω1  ℓ are the upper triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This  motivates the definition of the congruence subgroup Γ0(ℓ) ⊂ SL2(Z) defined as  Γ0(ℓ)  def  =  ��a  b  c  d  �  ∈ SL2(Z)  ���� c ≡ 0  mod ℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Namely, this is the group of elements of SL2(Z) which induce an automorphism of E [ℓ] fixing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Exercise 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Prove that the canonical map  SL2(Z) −→ SL2(Fℓ)  given by the reduction of the coefficients modulo ℓ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' To do it:  (a) Prove that an element of SL2(Fℓ) has a lift  �  a  b  c  d  �  with a, b, c, d ∈ Z such that a, b are nonzero and  prime to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  (b) Prove that for such a lift, c, d can be replaced by c′, d′ such that c ≡ c′ mod ℓ and d ≡ d′ mod ℓ so  that det  �a  b  c′  d′  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, the set of ℓ–isogenies between elliptic curves up to isomorphism is in one-to-one correspon-  dence with the complex variety  Γ0(ℓ)\\H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This variety has a compactification  X0(ℓ)  def  = Γ0(ℓ)\\H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This is a compact Riemann surface and it can be proved that such an object is actually algebraic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  is biholomorphic with a smooth complex projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This structure of algebraic curve is discussed  further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  27  The next statement gives a crucial information, namely the genus of X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — For a prime number ℓ > 3, the genus gℓ of X0(ℓ) equals  gℓ =  �  �  �  �  �  �  �  ℓ−1  12 − 1  if  ℓ ≡ 1  mod [12]  ℓ−5  12  if  ℓ ≡ 5  mod [12]  ℓ−7  12  if  ℓ ≡ 7  mod [12]  ℓ+1  12  if  ℓ ≡ 11  mod [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  We first need two technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Lemma 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let ℓ be a prime integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let Λ ⊆ C be a lattice and Λ1, Λ2 be two distinct lattices both  containing Λ and ♯(Λ1/Λ) = ♯(Λ2/Λ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Suppose that aΛ1 = Λ2 for some a ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then |a| = 1 and  aΛ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Equivalently, given an elliptic curve E over C and two distinct subgroups C1, C2 of cardinality  ℓ of E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' If the curves E /C1 and E /C2 are isomorphic, then there is an automorphism of E sending C1  onto C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Note that if E has such an automorphism, then it should be one of the two curves  mentioned in Theorem 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof of Lemma 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An adapted basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We claim that there exists ω1, ω2 ∈ C such that  (21)  Λ = Zω1 ⊕ Zω2  and  Λ1 = Zω1  ℓ ⊕ Zω2  and  Λ2 = Zω1 ⊕ Zω2  ℓ ·  The existence of ω1, ω2 can be obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' First, the structure theorem for finitely generated  modules over principal ideal rings asserts the existence of a basis η1, η2 such that  Λ = Zη1 ⊕ Zη2  and  Λ1 = Zη1  ℓ ⊕ Zη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Next, we claim that  Λ2 = Zη1 ⊕ Zuη1 + η2  ℓ  ,  for some u ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , ℓ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, consider  � 1  ℓ Λ  �  /Λ, which isomorphic to Fℓ × Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this quotient,  Λ1/Λ and Λ2/Λ are identified to two Fℓ–subspaces of dimension 1 in direct sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The subspace Λ1/Λ  is spanned by the class of η1  ℓ and the fact that Λ1/Λ and Λ2/Λ are in direct sum in  � 1  ℓ Λ  �  /Λ entails  that Λ2/Λ should be spanned by the class of uη1+η2  ℓ  for some u ∈ Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This implies that there exists  u ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' , ℓ − 1} such that uη1+η2  ℓ  ∈ Λ2 and hence  Zη1 ⊕ Zuη1 + η2  ℓ  ⊆ Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Then, since ♯(Λ2/Λ) = ℓ, we can deduce that the above inclusion is actually an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Finally, define  ω1, ω2 as  �ω1  ω2  �  def  =  �1  u  0  1  � �η1  η2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Note that the above change of variables is given by a matrix in SL2(Z) and provides a basis for Λ which  satisfies (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The modulus of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A classical notion in lattice theory is that of the determinant or volume  of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' It can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider Λ = Zω1 ⊕ Zω2 and regard C as a 2–dimensional  R–vector space with canonical basis (1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since any basis of Λ can be deduced from (ω1, ω2) by applying  a matrix in GL2(Z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' a matrix with determinant ±1, the quantity | det(ω1, ω2)| is the same for any  basis of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Hence we denote this quantity | det Λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From (21), we have  (22)  det Λ1 = 1  ℓ det Λ = det Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Moreover, the multiplication by a map z �→ az regarded as an R–linear endomorphism of C has determi-  nant |a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, writing a = a0 + DA1, the map is represented in the basis (1, i) by the matrix  �a0  −a1  a1  a0  �  ,  28  ALAIN COUVREUR  whose determinant is a2  0 + a2  1 = |a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, the assumption Λ2 = aΛ1 together with (22) give  det Λ1 = det Λ2 = |a|2 det Λ1,  which yields |a|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We aim to prove that aΛ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Suppose it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since Λ ⊆ Λ1, Λ ⊆ Λ2 and aΛ ⊆ aΛ1 = Λ2,  then Λ + aΛ ⊆ Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Recall that ♯Λ2/Λ = ℓ and ℓ is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, since we assumed that Λ ⊊ Λ + aΛ, we  get Λ+aΛ = Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Similarly, one deduces that a−1Λ+Λ = Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, from (21), we see that Λ1 +Λ2 = 1  ℓ Λ  and hence  a−1Λ + Λ + aΛ = 1  ℓ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  By induction,  a−sΛ + · · · + a−1Λ + Λ + aΛ + · · · + asΛ = 1  ℓs Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, for any s ⩾ 0, there exists a finite sequence (µs  i)s  i=−s of elements of Λ such that  s  �  i=−s  aiµs  i = 1  ℓs ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Then, for any N ⩾ 0,  N  �  s=0  s  �  i=−s  aiµs  i  =  N  �  s=0  1  ℓs ω1,  N  �  i=−N  aiνi  =  N  �  s=0  1  ℓs ω1,  where the νi’s are in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' When N goes to infinity, the right hand side is a convergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, so  does the left hand side and hence, its general term should go to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From the previous step, we know that  |a| = 1 and since the νi’s are in Λ which is discrete, then νi = 0 for any sufficiently large i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore,  the sequence of partial sums of the left–hand side is stationary while that of the right–hand side is note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore aΛ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Lemma 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let Ei  def  = C/(Z ⊕ Zi) and consider its automorphism group Gi induced by the multipli-  cations by {±1, ±i} in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then, any P ∈ Ei(C) which has a non trivial stabiliser under the action of Gi  is in E [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Similarly, let Eρ  def  = C/(Z ⊕ Zρ), where ρ = e  iπ  3  with its automorphism group Gρ induced by the  multiplications by {±1, ±ρ, ±ρ2}, then any P ∈ Eρ(C) with a non trivial stabiliser under the action of  Gρ is in E [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — In the case Ei, denote by Λi  def  = Z ⊕ Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Since Gi is cyclic of order 4, its only nontrivial  subgroups are {±1} and Gi itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A point P ∈ Ei(C) stabilised by {±1} corresponds to z ∈ C such that  z ≡ −z mod Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' That is to say 2z ∈ Λi and hence P ∈ Ei[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Similarly if P is stabilised by all Gi it is a  fortiori stabilised by {±1} and hence should be in Ei[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Consider now the case of Eρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Denote by Λρ  def  = Z ⊕ Zρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since Gρ is cyclic of order 6, its only possible  nontrivial subgroups are {±1}, {1, ρ2, ρ4} and Gρ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let us consider points which are stabilised by  one of these groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let P ∈ Eρ(C) stabilised by {±1}, then the very same reasoning as for Ei yields P ∈ Eρ[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Let P ∈ Eρ(C) stabilised by {1, ρ2, ρ4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This corresponds to z ∈ C satisfying ρ2z ≡ z mod Λρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Writing z = a + bρ2 for some a, b ∈ R and using the relation 1 + ρ2 + ρ4 = 0, we get  a + ρ2b ≡ −b + ρ2(a − b)  mod Λρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  This entails that  �  −b  =  a + µ  a − b  =  b + ν,  CODES AND MODULAR CURVES  29  where µ, ν ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' By elimination, we deduce that 3a ∈ Z and 3b ∈ Z, that is to say z ∈ 1  3Λρ and hence  P ∈ Eρ[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, the previous discussion entails that a point stabilised by the whole Gρ should be in Eρ[2]∩Eρ[3],  and hence is nothing but OEρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof of Theorem 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The idea is to consider the projection map π : X0(ℓ) → X0(1), which sends a  class of isomorphisms of isogenies φ : E → E ′ onto the isomorphism class of E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This map is algebraic (this  will appear more naturally in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The objective is to apply Riemann Hurwitz formula (Theorem 31)  to π in order to compute the genus of X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The degree of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The degree of π is the generic number of pre-images of a point of X0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such  a point corresponds to a curve E up to isomorphism and its pre-image is the set of isomorphism classes  of ℓ–isogenies E → E ′ or equivalently, the ismomorphism classes of pairs (E , C) where C is a subgroup of  cardinality ℓ of E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Lemma 75, if E has no nontrivial automorphism, then two distinct subgroups  C1, C2 provide non isomorphic pairs (E , C1), (E , C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, in this situation, the number of pre-images  of E by π corresponds to the number of subgroups of cardinality ℓ in E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since ℓ is prime, then from  Theorem 54, E [ℓ] ≃ Fℓ × Fℓ and hence is a vector space of dimension 2 over Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Next, a subgroup of  cardinality ℓ of E [ℓ] is nothing but a subspace of dimension 1 and the number of subspaces of dimension  1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' of lines) of Fℓ × Fℓ equals ♯P1(Fℓ) = ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus,  deg π = ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Now, the map is ramified at the points corresponding to curves with nontrivial automorphisms and  possibly the point at inifinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Theorem 69, the curves with nontrivial automorphisms correspond  to the tori C/(Z ⊕ Zi) and C/(Z ⊕ Zρ), where ρ = e  iπ  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For these tori, we need to understand the action  of the automorphisms on the ℓ–torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Ramification at C/(Z ⊕ Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The curve is equipped with a nontrivial automorphism η of  order 4, which corresponds to the multiplication by i in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This automorphism acts on the ℓ–torsion  and, from [Sil09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='8], such an automorphism is a group automorphism and hence its action  on E [ℓ] regarded as an Fℓ–vector space is Fℓ–linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We denote by ηℓ, the automorphism η restricted to  E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Lemma 77, since ℓ > 3 any point in E [ℓ] \\ {OE } has has an orbit of cardinality 4 under ηℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, ηℓ has order 4 and two situations may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Either ℓ ≡ 1 mod 4, then Fℓ contains fourth  roots of 1 and ηℓ regarded as an Fℓ–automorphism of Fℓ × Fℓ is diagonalisable as  �ι  0  0  −ι  �  ,  where ι denotes a primitive fourth root of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this situation, ηℓ acts on the lines of Fℓ × Fℓ by fixing the  two lines corresponding to the eigenspaces of ηℓ and any other line has an orbit of cardinality 2, indeed  η2  ℓ = −Id, which leaves any line invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Two lines of E [ℓ] in a same orbit under ηℓ correspond to a  same point in X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This point is a ramification point with ramification index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, if ℓ ≡ 1  mod 4, then there are 2 unramified points in the pre-image of the isomorphism class of E by π and ℓ−1  2  ramified points with ramification index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Otherwise ℓ ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this situation ηℓ has no eigenspace in E [ℓ] and the orbit of any line has  cardinality 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, there are ℓ+1  2  points above E which are all ramified with ramification index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Ramification at C/(Z ⊕ Zρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Here we have an automorphism η of order 6 and denote again  by ηℓ its restriction to E [ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Here again, from Lemma 77, we know that ηℓ has also order 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In this  situation, if ℓ ≡ 1 mod 3, then Fℓ contains sixth roots of unity and ηℓ is diagonalisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the  two eigenspaces of ηℓ are left invariant and any other Fℓ–line of E [ℓ] has an orbit of cardinality 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed,  here again ρ3 = −Id and hence leaves any line globally invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In such a situation, the pre-image of  E consists in 2 unramified points corresponding to the two eigenspaces of ηℓ in E [ℓ] and ℓ−1  3  points with  ramification index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  If ℓ ≡ 2 mod 3, then any line of E [ℓ] has an orbit of cardinality 3 under the action of ηℓ and hence  the pre-image of E by π consists in ℓ+1  3  points, all with ramification index 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  30  ALAIN COUVREUR  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Ramification at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The point at infinity of X0(1) is the quotient of P1(Q) under the  action of SL2(Z), which, from Proposition 71, consists in a single orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We wish to estimate the number  of orbits in P1(Q) under the action of Γ0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We claim that their number is 2, namely, the orbit of (0 : 1)  and that of (1 : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed,  Γ0(ℓ) · (0 : 1) = {(b : d) ∈ P1(Q) with gcd(b, d) = 1 and d prime to ℓ}  and  Γ0(ℓ) · (1 : 0) = {(a : c) ∈ P1(Q) with gcd(a, c) = 1 and ℓ dividing c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  One easily sees that the two orbits form a partition of P1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This entails that the pre-image of the point  at infinity of X0(1) consists in two points P, Q whose ramification indexes satisfy eP + eQ = ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Computation of the genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Denote by gℓ the genus of X0(ℓ) and by g1 = 0 that of  X0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Riemann–Hurwitz formula asserts that  2gℓ − 2 = (2g1 − 2)(ℓ + 1) + νi + νρ + ν∞,  where νi, νρ and ν∞ are the respective contributions of the ramifications above C/(Z ⊕ Zi), C/(Z ⊕ Zρ)  and the point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We get  ν2 =  �  ℓ−1  2  if  ℓ ≡ 1  mod 4  ℓ+1  2  if  ℓ ≡ 3  mod 4 ,  ν3 =  � 2 ℓ−1  3  if  ℓ ≡ 1  mod 3  2 ℓ+1  3  if  ℓ ≡ 2  mod 3  and  ν∞ = ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  An easy but cumbersome calculation treating separately the four cases ℓ ≡ 1, 5, 7, 11 mod 12 yields the  expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The modular equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — To conclude this section, we give a statement whose proof is omitted  but which may help the reader to be convinced that X0(ℓ) has a structure of algebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We refer  the reader to [Mil17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1] for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — There exists an irreducible polynomial Φℓ ∈ Z[x, y] such that for any pair E , E ′ of  elliptic curves related with a degree ℓ isogeny E → E ′, then Φℓ(j(E ), j(E ′)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Remark 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — A database of the polynomials Φℓ for small values of ℓ is available on Andrew Suther-  land’s webpage: https://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='edu/∼drew/ClassicalModPolys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='html  Let us give some comments about this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' First, note that the projective closure of the complex  curve of equation Φℓ(x, y) = 0 is a “singular model” for X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Indeed, any point of X0(ℓ) corresponds to  an isomorphism class of ℓ–isogeny E → E ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' This yields a rational map  �  X0(ℓ)  −→  P2(C)  (E → E ′)  �−→  (j(E ) : j(E ′) : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The image of this map is contained into the curve with equation Φℓ(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' However, this latter curve  is full of singularities and hence is not isomorphic to X0(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Nevertheless, (and this is far from being  obvious) this permits to deduce that X0(ℓ) is itself defined over Q and hence, its reduction modulo p  makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  An interesting fact is that, since Φℓ ∈ Z[x, y], for any pair of ℓ–isogenous curves E → E ′ over Fp  we have Φℓ(j(E ), j(E ′)) ≡ 0 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, if ℓ and p are prime to each other, it is known that  the polynomial Φℓ is irreducible modulo p (see for instance [Mor90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, the curve over  Fp of equation Φℓ(x, y) = 0 turns out to be a singular model of a smooth curve over Fp, that we will  also denote by X0(ℓ) such that X0(ℓ)(Fp) parameterises ℓ–isogenies E → E ′ over Fp up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  These curves over Fp will be the objects of interest in order to prove the main theorem of this course,  namely Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Finally, the reader interested in a rigorous study of the reductions of modular curves cannot avoid the  language of schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For such a development, we refer the reader to the article of Celgene and Rapoport  [DR73] or the book of Katz and Mazur [KM85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  CODES AND MODULAR CURVES  31  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Proof of the main Theorem  Now, we almost have the material to prove Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We have our family of curves X0(ℓ) for ℓ a  prime integer distinct from the characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Genus of modular curves over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let us briefly discuss the genus of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  The discussion to follow is far from being trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Thus, the reader is encouraged first to directly admit  the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Namely that the genus of a modular curve over a finite field is that of its complex  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Let us briefly sketch the reasons why this holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  It is known (see for instance in [Mor90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9]) that, the curve X0(ℓ) has a smooth projective  model described by equation with coefficients in Z and whose reduction modulo p is smooth too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Next, as already mentioned in Remark 29, two different notions of genus are associated to a curve,  the arithmetic genus pa and the geometric one g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The genus introduced by Definition 25 in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='7 is the  geometric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The arithmetic genus, which can be defined for instance from the Hilbert function of the  variety [Har77, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' IV], is always larger than or equal to the geometric one and they coincide if and  only if the curve is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Next, Grauert Theorem [Har77, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='9] permits to assert that the reduction modulo p of the  aforementioned model X0(ℓ) has the same arithmetic genus as the complex curve itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreover, since  this model and its reduction are smooth, they also have the same geometric genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the genus of  the curve X0(ℓ) over Fp is the same as that of its complex counterpart and hence is given by Theorem 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The locus of supersingular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — There remains to get an estimate of the number of  rational points of such curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' For this we will focus on Fp2–points since their number can be bounded  from below using the two following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proposition 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Let E be a supersingular elliptic curve over Fp, then E is defined over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — By definition, a supersingular curve E satisfies E [p] = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the multiplication by p  map [p] : E → E is totally inseparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Consider now the Frobenius map  φ :  �  E  −→  E (p)  (x, y)  �−→  (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  It is a degree p isogeny, hence it has a dual isogeny ˆφ such that ˆφ◦φ = [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Since [p] is totally inseparable,  ˆφ should be inseparable either and hence, so should be the Frobenius map  ˆφ :  �  E (p)  −→  E (p2)  (x, y)  �−→  (xp, yp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Thus, E (p2) = E and hence E is defined over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Theorem 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — The number of Fp–isomorphism classes of supersingular elliptic curves over Fp equals  � p  12  �  +  �  �  �  �  �  �  �  0  if  p  ≡  1  mod 12  1  if  p  ≡  5  mod 12  1  if  p  ≡  7  mod 12  2  if  p  ≡  11  mod 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — See [Sil09, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — With these two last statements at hand we can finally provide  the proof of Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' We restrict the proof to the case p ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Note that Tsfasman–Vlăduţ–Zink  theorem remains true when p = 2, 3 but the coding theoretic interest is rather limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Proof of Theorem 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' — Consider the sequence of curves X0(ℓ) for ℓ ≡ 11 mod 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Theorem 74  it has genus gℓ = ℓ+1  12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Theorem 81, the curve X0(1) has at least p−1  12  Fp2–rational points corre-  sponding to isomorphism classes of supersingular elliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Such an elliptic curve with no nontrivial  automorphism has ℓ + 1 pre-images in X0(ℓ) which also correspond to supersingular elliptic curves and  32  ALAIN COUVREUR  hence are Fp2–rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Depending on the class of p modulo 12 the curves with j–invariant 0 and  1728 may be supersingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' More precisely, from [Sil09, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4 & V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='5],  for p ≡ 1 mod 12, both curves are ordinary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' non supersingular) and then any supersingular  curve has no nontrivial automorphism and hence has ℓ + 1 pre-images in X0(ℓ)(Fp2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore,  from Theorem 81,  ♯X0(ℓ)(Fp2) ⩾ (ℓ + 1)p − 1  12 ·  for p ≡ 5 mod 12, the curve with j = 0 is supersingular and the one with j = 1728 is ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Therefore, there are p−5  12 + 1 supersingular curves and all of them but one have ℓ + 1 distinct pre-  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The remaining curve is the one with j = 0 and an automorphism group of order 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Its  treatment is very similar to the proof of Theorem 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consider the action of the automorphism of  order 6 on the ℓ–torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' From Lemma 77, this induces an automorphism η or order 6 of E [ℓ] ≃ F2  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Since ℓ ≡ 11 mod 12, then ℓ ≡ 2 mod 3 and hence Fℓ does not contains the sixth roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Thus, η is not diagonalisable and hence cannot fix a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Consequently, one proves that the orbit  of any line is the union of 3 distinct lines (η3 = −Id, which fixes the lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Therefore, the curve  with j–invariant 0 has ℓ+1  3  pre-images and Consequently, using Theorem 81:  ♯X0(ℓ)(Fp2) ⩾ (ℓ + 1)p − 5  12  + ℓ + 1  3  = (ℓ + 1)p − 1  12 ·  for p ≡ 7 mod 12, the curve with j = 0 is ordinary and the one with j = 1728 is supersingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A  similar reasoning yields  ♯X0(ℓ)(Fp2) ⩾ (ℓ + 1)p − 7  12  + ℓ + 1  2  = (ℓ + 1)p − 1  12 ·  for p ≡ 11 mod 12, both curves are supersingular and we get:  ♯X0(ℓ)(Fp2) ⩾ (ℓ + 1)p − 11  12  + ℓ + 1  2  + ℓ + 1  3  = (ℓ + 1)p − 1  12 ·  In summary, we always have a lower bound (ℓ + 1) p−1  12 on the number of rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Then  ♯X0(ℓ)(Fp2)  gℓ  ⩾  p−1  12 (ℓ + 1)  � ℓ+1  12  �  = p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Thus, over Fq for q = p2, we identified a family of curves whose number of Fq–rational points goes to  infinity and whose ratio, number of Fq–points divided by the genus goes to √q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Which turns out to  be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  References  [BH08]  Peter Beelen and Tom Høholdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The decoding of algebraic geometry codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In Advances in algebraic  geometry codes, volume 5 of Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Coding Theory Cryptol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', pages 49–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', Hackensack,  NJ, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Cou16]  Alain Couvreur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Introduction to coding theory, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Personal lecture notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [CR21]  Alain Couvreur and Hugues Randriambololona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometry codes and some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Cary Huffman, Jon-Lark Kim, and Patrick Solé, editors, Concise Encyclopedia of Coding Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Chapman and Hall/CRC, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [DR73]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Deligne and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Les schémas de modules de courbes elliptiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' 349, pages 143–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Springer, Berlin, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' Algebraic geometry codes: general theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In Advances in algebraic geometry codes,  volume 5 of Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Coding Theory Cryptol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=', Hackensack, NJ, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' Advanced Book Classics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Addison-Wesley Publishing Company Ad-  vanced Book Program, Redwood City, CA, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An introduction to algebraic geometry, Notes written  with the collaboration of Richard Weiss, Reprint of 1969 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Gop81]  Valerii D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' Codes on algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' Nauk SSSR, 259(6):1289–1290, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In  Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content=' A tower of Artin-Schreier extensions of function fields at-  taining the Drinfeld-Vladut bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Inventiones Mathematicae, 121:211–222, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Har77]  Robin Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometry, volume 52 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Springer-Verlag,  New York, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [HP95]  Tom Høholdt and Ruud Pellikaan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' On the decoding of algebraic–geometric codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='  Theory, 41(6):1589–1614, Nov 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [HvLP98] Tom Høholdt, Jacobus Hendricus van Lint, and Ruud Pellikaan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometry of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In  Handbook of coding theory, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' I, II, pages 871–961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' North-Holland, Amsterdam, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Iha81]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Ihara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Some remarks on the number of rational points of algebraic curves over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Tokyo Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' IA Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', 28:721–724, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [KM85]  Nicholas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Katz and Barry Mazur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Arithmetic moduli of elliptic curves, volume 108 of Annals of  Mathematics Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Princeton University Press, Princeton, NJ, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Lor96]  Dino Lorenzini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' An invitation to arithmetic geometry, volume 9 of Graduate Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  American Mathematical Society, Providence, RI, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [LPS88]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Lubotzky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Phillips, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Sarnak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Ramanujan graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Combinatorica, 8(3):261–277, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Mar88]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Margulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Explicit group-theoretic constructions of combinatorial schemes and their applications  in the construction of expanders and concentrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Problemy Peredachi Informatsii, 24(1):51–60, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Mil17]  James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Milne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Modular functions and modular forms (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='31), 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Available at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='jmilne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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+page_content='  [Mor90]  Carlos J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Moreno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic curves over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Cambridge tracts in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Cambridge  University Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Sha94]  Igor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Shafarevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Basic algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Springer-Verlag, Berlin, second edition, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Sil09]  Joseph Silverman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' The arithmetic of elliptic curves, volume 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Springer-Verlag New York, second  edition, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Ste99]  Serguei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Stepanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Codes on algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Kluwer Academic/Plenum Publishers, New York, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Sti09]  Henning Stichtenoth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic function fields and codes, volume 254 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Springer-Verlag, Berlin, second edition, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [TVN07]  Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Tsfasman, Serge Vlăduţ, and Dmitry Nogin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Algebraic geometric codes: basic notions,  volume 139 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' American Mathematical Society, Providence,  RI, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [TVZ82]  Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Tsfasman, Serge G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Vlăduţ, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Zink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Modular curves, Shimura curves, and Goppa  codes, better than Varshamov-Gilbert bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', 109:21–28, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [VD83]  Sergei G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Vlăduţ and Vladimir G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Drinfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Number of points of an algebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=',  17:53–54, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [VM84]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Vlăduţ and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Manin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Linear codes and modular curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' In Current problems in mathematics,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' 25, Itogi Nauki i Tekhniki, pages 209–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Nauk SSSR Vsesoyuz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Nauchn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' i Tekhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=', Moscow, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  [Wal00]  Judy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Codes and curves, volume 7 of Student Mathematical Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' American Mathematical  Society, Providence, RI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' Institute for Advanced Study (IAS), Princeton, NJ, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content=' IAS/Park City  Mathematical Subseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='  Alain Couvreur, Inria, France E-mail : alain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
+page_content='couvreur@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQf-gZN/content/2301.03569v1.pdf'}
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