diff --git "a/EtE1T4oBgHgl3EQf-gZN/content/tmp_files/2301.03569v1.pdf.txt" "b/EtE1T4oBgHgl3EQf-gZN/content/tmp_files/2301.03569v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/EtE1T4oBgHgl3EQf-gZN/content/tmp_files/2301.03569v1.pdf.txt" @@ -0,0 +1,2359 @@ +CODES AND MODULAR CURVES +by +Alain Couvreur +Abstract. — 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. 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. +Contents +Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1. Linear Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2. Algebraic curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3. Algebraic geometry codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 +4. Elliptic curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 +5. Modular curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 +6. Proof of the main Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 +References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 +Introduction +Algebraic Geometry (AG) codes is a particularly exciting topic lying at the intersection between +number theory, algebraic geometry and coding theory. 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. 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.s. genus. 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. This remarkable and totally unexpected result turned out to be the first stone +of the development of a whole theory: that of AG codes. Surprisingly, a very comparable breakthrough +happened in graph theory the late 80’s. 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.e. the length of their shortest cycle, exceeds the girth obtained with the +probabilistic method. In both situations, coding theory and graph theory, the use of elegant algebraic +structures unexpectedly beat random constructions. +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. 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:2301.03569v1 [math.NT] 9 Jan 2023 + +2 +ALAIN COUVREUR +so-called Drinfeld–Vlăduţ [VD83]. 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. +• 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. al. and Ihara provide an enough +detailed proof. In both articles, the proof is made of less than ten lines hiding a huge amount of +prerequisites. +• 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. +Outline of these notes. — We start in Section 1 with bases on linear codes and their asymptotic +behaviour. Section 2 gives an introduction to algebraic curves by providing the necessary material in +algebraic geometry. Section 3 introduces algebraic geometry codes and states the main result: Tsfasman– +Vlăduţ–Zink Theorem. The remainder of the notes are dedicated to the proof of this statement. Sections 4 +and 5 provide further material on elliptic and modular curves respectively. Section 6 concludes the proof. +Acknowledgements. — 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. I would probably never have found the time to do +it if they did not ask me for. Several colleagues spent time to carefully read these notes. 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. +The author is funded by the french Agence nationale de la recherche for the collaborative project +ANR-21-CE39-0009-BARRACUDA. +1. Linear Codes +1.1. Context. — In the sequel we are interested in linear q–ary codes, which are linear subspaces of Fn +q . +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. +The Hamming distance between two vectors x, y ∈ Fn +q is denoted by +dH(x, y) +def += ♯{i ∈ {1, . . . , n} | xi ̸= yi}. +The Hamming weight of a vector is its Hamming distance to the zero vector. +∀x ∈ Fn +q , +wH(x) +def += dH(x, 0). +1.2. Linear codes. — Unless otherwise specified, a code will denote a linear subspace C ⊆ Fn +q . The +vectors of C are usually referred to as codewords. 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. 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.uzh.ch/act/ + +CODES AND MODULAR CURVES +3 +subscript. 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? +A code will be considered as “good” if both k and d are as close as possible to n. +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. +The rationale behind this question is that both k and d quantify some feature of linear codes. 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. The rate is nothing +but the ratio of information divided by the quantity of data which is actually sent across the channel. +Hence, the rate R = k/n quantifies the efficiency of encoding. +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). +Finally, suppose that our objective is to correct errors from a given channel. 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, . . . , 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. The average weight +of our error vector satisfies +E(wH(e)) = pn. +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. 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. +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. +Remark 1. — 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. +In this setting, the question of the achievable pairs (δ, R) ∈ [0, 1] × [0, 1] remains open. Some bounds +are known: +• Singleton bound immediately entails that R + δ ⩽ 1; +• A more precise bound called Plotkin bound entails that R + δ ⩽ 1 − 1 +q. See for instance [Cou16, +Chap. 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. +Exercise 2. — Prove this last assertion. +• 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). 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. An upper bound for +αq is given by the minimum of all the known upper bounds on the achievable pairs (δ, R). +(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. 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. +In short, the pair (δ, R) for a random sequence satisfies R = 1 − Hq(δ). +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. 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. A breakthrough is due to Tsfasman, Vlăduţ and Zink [TVZ82] who +showed that the asymptotic Gilbert Varshamov bound is not always optimal. More precisely, they proved +the following statement. +Theorem 3. — Let q = p2 where p is a prime number. 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. — Actually, the result holds for any q = p2m where p is prime and m ⩾ 1. +Remark 5. — 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). This statement on curves has proved +by Tsfasman, Vlăduţ and Zink in [TVZ82] and independently by Ihara in [Iha81]. However, Ihara did +not rely this result with coding theory while Tsfasman et. al. did. +It turns out that, as illustrated by Figures 1 and 2, for q ⩾ 49, such codes beat Gilbert Varshamov +bound. These codes, are actually far from being random and are constructed using elegant techniques +from number theory and algebraic geometry. 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. +2. 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. In particular, +here most of the proofs will be omitted. 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; +• for a reader who feels comfortable with the language of schemes, the contents of this section could +appear to be dirty. +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. +(4)This can be formalised as follows, consider the set of all codes of length n and dimension Rn in Fn +q . This set is finite, +and let C be a random variable uniformly distributed over this set. + +CODES AND MODULAR CURVES +5 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +R +δ +GV bound +TVZ bound +Figure 1. The TVZ bound for q = 49 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +R +δ +GV bound +TVZ bound +Figure 2. 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. 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. + +6 +ALAIN COUVREUR +• A more advanced presentation on algebraic geometry codes appears in Tsfasman Vlăduţ and Nogin’s +book [TVN07] and Stepanov [Ste99] . +• 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]. Lorenzini’s book [Lor96] +can be an excellent reading either if you wish a better focus on the arithmetic side. +2.1. Plane curves and functions. — Let K be a perfect field(5) and K be its algebraic closure. We +denote by A2(K) and P2(K) respectively the affine and projective planes over K. 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]. +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]. 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. +Such an affine (resp. projective) curve is said to be irreducible if f (resp. F) is an irreducible polynomial +in K[x, y] (resp. K[X, Y, Z]) and absolutely irreducible if f (resp. F) is irreducible when regarded as an +element of K[x, y] (resp. K[X, Y, Z]). +Example 6. — Suppose K = Q and consider the affine curve X with equation x2 − 2y2 = 0. This +curve is irreducible but not absolutely irreducible. 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. Geometrically speaking, X is the union of the two lines with respective equations +x − +√ +2y = 0 and x + +√ +2y = 0. These lines are not defined over Q but their union is. +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 . 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). +For an affine curve X , elements of K[x, y]/(f) can be understood as restrictions of polynomial functions +to the curve X . 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, § 1.7]) 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. +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 +×. 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). Note however that, for such a +polynomial, vanishing at a point is a well–defined notion. 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. +Remark 7. — Note that we are juggling with K and K. 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. 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. + +CODES AND MODULAR CURVES +7 +curve. 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! +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. This explains the rationale behind considering the points with coordinates in K. +Remark 8. — Note that when speaking about functions, these objects may not be defined everywhere +on the curve and may have some poles somewhere. These objects can be understood as the algebraic +geometric counterpart of meromorphic functions in complex analysis. +Remark 9. — It is well–known that the projective plane can be covered by affine planes sometimes +called affine charts. 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). +The images of these three embeddings cover the full projective plane. 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. 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. Hence, one can deduce affine curves (affine charts) from a given projective curve. 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 . +A crucial fact is that a curve and its projective closure share a common object : their function field +remains the very same one. +2.2. Points. — A point of X is an element (a, b) ∈ A2(K) (resp. (u : v : w) ∈ P2(K)) such that +f(a, b) = 0 (resp. F(u, v, w) = 0). A point is said to be a rational point or a K–point if its coordinates +all lie in K. More generally, given an extension L/K, one can define the notions of L–points of X . The +set of K–points or L–points of X respectively denoted by X (K) and X (L). One of topics of interest +for us in the sequel is the case K = Fq. In this situation, one sees easily that X (Fq) is finite. Indeed, it +is a subset of A2(Fq) or P2(Fq) which are both finite sets. 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. +Given an affine (resp. projective) curve X defined by the equation f(x, y) = 0 (resp. F(X, Y, Z) = 0) +over a field K, a point P ∈ X (K) with coordinates (xP , yP ) (resp. (uP : vP : wP )) is said to be singular +if +∂f +∂x(xP , yP ) = ∂f +∂y (xP , yP ) = 0 +resp. +∂F +∂X (uP , vP , wP ) = ∂F +∂Y (uP , vP , wP ) = ∂F +∂Z (uP , vP , wP ) = 0. +A non singular point is said to be regular. A curve without singular points is said to be regular or smooth. +On the other hand a curve having at least one singular point is said to be singular. It can be proved that +the set of singular points of a curve is always finite. +From now on, unless otherwise specified, any curve is smooth projective and absolutely irre- +ducible. +2.3. Galois action on points. — Recall that, for the sake of simplicity, we restrict the definitions to +the case where the base field K is perfect. This is not a strong restriction for the subsequent purpose +where K will always be either finite or of characteristic zero. +Given a curve X defined over K, any point P ∈ X (K) has coordinates (xP , yP ) (or (uP : vP : wP ) in +the projective setting). 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 . 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). 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. +Definition 10. — 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). +The number of elements in such an orbit is referred to as the degree of the closed point. It is also +the extension degree [K(P) : K]. 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. +Remark 11. — 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 ). +Example 12. — Consider the case K = Q and the affine curve C with equation x2 + y2 − 1 = 0 +(a circle). The point with coordinates (1, 0) is a rational point of X , i.e. an element of C (Q). The +complex point (2, +√ +3i), (where i2 = −1), is a geometric point of C , i.e. an element of C (C). Finally, +{(2, i +√ +3), (2, −i +√ +3)} is a closed point of degree 2 of C . +2.4. Maps between curves. — As usually in algebra, once structures have been introduced: for +instance groups, rings, modules, etc., one introduces morphisms between these objects. In the case of +curves, we are interested in two kinds of maps referred to as morphisms and rational maps. A rational +map between two affine (resp. projective) curves X , Y contained in A2 (resp. P2) is a map: +ϕ : +� +X +��� +Y +(x, y) +���→ +(ϕ1(x, y), ϕ2(x, y)) +resp. +ψ : +� +X +��� +Y +(u : v : w) +�−→ +(ψ1(u, v, w), ψ2(u, v, w), ψ3(u, v, w)). +where φ1, φ2 (resp. 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). The dashed arrow ��� is here to emphasize the fact that this +map is not defined at every point but only on a subset(6). For affine curves, at any point where ϕ1, ϕ2 +have no pole, the map is defined and said to be regular. 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. A rational map between two curves X ��� Y is said to be +regular if it is regular at any point of X . +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 ). +The degree of ϕ is defined as the degree of this field extension. +Example 13. — Back to example 12. The map +(2) +� +C +��� +P1 +(x, y) +�−→ +(x : 1) +is a rational map. It is also possible to construct a rational map P1 → C as +(3) +� +P1 +��� +C +(u : v) +�−→ +� +v2−u2 +u2+v2 , +2uv +u2+v2 +� +. +Note that these two maps are not inverses to each other. +Finally, the following statements are well–known. Their proofs are omitted. +(6)This subset turns out to be dense for a suitable topology called Zariski topology + +CODES AND MODULAR CURVES +9 +Proposition 14. — Let h : X +→ Y +be a rational map between two smooth projective absolutely +irreducible curves X , Y . +(i) if X is smooth, then h is regular; +(ii) if h is non constant, then it is surjective. +2.5. Valuations. — Recall that a local ring is a ring having a unique maximal ideal. The term local +comes precisely from the fact that many such rings may be understood as rings of functions characterized +by a local property. 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.e. have no +pole) at P. Namely +OX ,P +def += +�a(x, y) +b(x, y) ∈ K(X ) +��� b(xP , yP ) ̸= 0 +� +. +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 +� +. +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. Such a generator t of mX ,P is called a local parameter (or sometimes a uniformising parameter) +at P. 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). Next, one can easily prove that K(X ) is nothing but the field +of fractions of OX ,P . 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). +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); +• ∀a, b ∈ K(X ) \ {0}, vP (a + b) ⩾ min{vP (a), vP (b)} and equality holds when vP (a) ̸= vP (b). +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.e. the field of rational functions +on X with coefficients in K. Therefore, the valuation may be defined at any possible point. +2.6. Divisors. — 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. +Given a smooth curve X over a perfect field K, a (geometric) divisor is a formal Z–linear combination +of geometric points of X . A divisor is said to be rational if it is globally invariant under the action of +Gal(K/K). Equivalently, it is a formal sum of closed points of X . +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 . The set {P1, . . . , Pr} is referred to as +the support of G. The divisor is rational if for any i, j ∈ {1, . . . , r} such that Pi, Pj are in the same orbit +under the action of Gal(K/K), then ni = nj. +Remark 15. — We emphasize that a sum of rational points yields a rational divisor but the converse +is false. A rational divisor may be a sum of non rational points. See the subsequent Example 16. +Example 16. — 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). Then, aP + bP ′ + cQ is a rational divisor on the +curve if and only if a = b. + +10 +ALAIN COUVREUR +The group of divisors is equipped with a partial order relation denoted ⩽ and defined as follows. 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 . +In particular, a divisor G is said to be positive if G ⩾ 0, where 0 denotes the zero divisor. +Given a divisor G as in (4), its degree is defined as +deg G +def += n1 + · · · + nr. +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. +Such a divisor is called a principal divisor. +Remark 17. — For such an object to be a divisor, we need to show that the sum (5) is finite, i.e. that +the nP ’s are all zero but a finite number of them. This is actually due to a well–known fact appearing in +the next statement whose proof is omitted. +Proposition 18. — A nonzero rational function on a curve has only a finite number of zeroes and +poles. +Remark 19. — It is worth noting that a principal divisor is rational. 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). +The following very classical statement is crucial in the sequel. +Proposition 20. — The degree of principal divisor is always 0. +We finish this discussion with a statement that we admit and which will be useful later. +Proposition 21. — A principal divisor (f) associated to f ∈ K(X )× is zero if and only if f is constant. +2.7. Genus and Riemann–Roch Theorem. — The most elementary curve one may define is the +affine line A1 and its projective closure being the projective line P1. Regular functions on A1 are nothing +but univariate polynomials. Regarding such a polynomial h(x) ∈ K[x] as rational function on P1, it has +a pole at the point “at infinity”, i.e. the points with homogeneous coordinates (1 : 0) and one can prove +that the valuation at this pole is nothing but − deg h. +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. 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∞. +As the following definition suggests, Riemann–Roch spaces are generalisations for curves of the spaces +K[x]⩽n. +Definition 22 (Riemann–Roch space). — Let X be a smooth projective absolutely irreducible +curve over K and G be a rational divisor on X. Then the Riemann–Roch space associated to G is defined +as +L(G) +def += {h ∈ K(X ) | (h) + G ⩾ 0} ∪ {0}. +This is a vector space over K. +Remark 23. — According to the previous discussion, on P1, we have L(nP∞) ≃ K[x]⩽n. + +CODES AND MODULAR CURVES +11 +The following statement summarises some properties of Riemann–Roch spaces. +Proposition 24. — +(i) A Riemann–Roch space is a vector space over K of finite dimension; +(ii) For any rational divisor G < 0, we have L(G) = {0}; +(iii) For any rational divisor G, we have dimK L(G) ⩽ deg G + 1. +With the above statement at hand, we can introduce a fundamental invariant of a curve: its genus. +There are dozens of manners to define this object but none of them is trivial. The one given in these +notes is far from being satisfying since it is clearly not intuitive. However, it permits to define the object +with a minimal amount of material. +Definition 25 (Genus of a curve). — Let X be a smooth projective absolutely irreducible curve. +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 . +Remark 26. — Proposition 24 (iii) asserts that the involved minimum exists and that the genus is +nonnegative. +Exercise 27. — Prove the statement of Remark 26. +Exercise 28. — Using Definition 25, prove that the genus of the projective line P1 is zero. +Note that the effective computation of the genus is not a simple task. However, for smooth plane +curves of degree d, there is a closed formula (see [Ful89, Prop. VIII.5]): +g = (d − 1)(d − 2) +2 +· +This permits in particular to prove that the projective line and smooth conics have genus 0. +Remark 29. — The notion of genus can actually be defined for singular curves. In this context, two +distinct invariants respectively called arithmetic genus and geometric genus can be defined. These two +invariants coincide when the curve is smooth. +We conclude this section with Riemann–Roch Theorem, which is a crucial statement in the theory of +algebraic curves. This statement is admitted and we refer the reader to Fulton [Ful89] or Stichtenoth’s +[Sti09] book for a proof. The first part of the statement is actually a straightforward consequence of the +definition we gave for the genus (Definition 25). +Theorem 30 (Riemann–Roch Theorem). — Let X be a smooth absolutely irreducible curve of +genus g over K and G be a rational divisor on X . Then +dimK L(G) ⩾ deg G + 1 − g +and equality holds when deg G > 2g − 2. +2.8. The Riemann–Hurwitz formula. — 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 . Recall that, according to Proposition 14, +such a map is regular and surjective. Denoting by δ its degree (see § 2.4 for the definition of degree), +consider any geometric point P ∈ Y (K). 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. +The finite number of points of Y (K) where this no longer holds are called ramified points. 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 ). It can be proved that this definition does not depend on the choice of +the local parameter t at P. According to the previous definition, for any point Q ∈ X (K) but finitely +many of them, we have eQ = 1. + +12 +ALAIN COUVREUR +Here we have the material to state Riemann–Hurwitz formula. +Theorem 31 (Riemann–Hurwitz formula (Tame version)). — Let X , Y be two smooth projective +absolutely irreducible curves over K and ϕ : X ��� Y be a rational map. Suppose that for any Q ∈ Y (K), +the ramification index eQ is prime to the characteristic of K. Then, the genera gX , gY of X , Y are related +by the following formula. +(2gX − 2) = deg ϕ · (2gY − 2) + +� +Q∈Y (K) +(eQ − 1). +Remark 32. — According to the previous discussion, the terms of the sum in the above formula are +all zero but a finite number of them. +Remark 33. — 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. See [Sti09, Thm. 3.4.13]. +This formula is particularly useful since many curves X are described by a morphism X → P1. 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. +Example 34. — Consider the map (2) of Example 13 but here we regard the curve C as a curve over +C. 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}. Therefore, there are two ramified points both with ramification index 2 (one can show that +the map does not ramify at infinity). Moreover, the map has degree 2. Then, Riemann–Hurwitz formula +yields +2gC − 2 = 2(2gP1 − 2) + 2. +Since gP1 = 0, we deduce that gC = 0 too. +2.9. What about non plane curves? — A last important fact is that some curves are not plane and +may be contained in PN for N > 2. 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. 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. +So, the question is... what remains true when considering curves in PN with N > 2? and actually, how +are such objects defined? +We define a projective subvariety of PN as the common vanishing locus of the elements of a homoge- +neous ideal I ⊆ K[X0, . . . , XN]. 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. Then, the dimension of the variety can be defined as the transcendence degree of the function field +over K. A curve will be a variety of dimension 1. Smoothness can be defined very similarly by requiring +a non simultaneous vanishing of all the partial derivatives with respect to the N + 1 variables. 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. Finally all the previous statements on plane curves actually hold +for any curve. +3. Algebraic geometry codes +Now, we have the necessary material to define algebraic geometry (AG) codes. Before, let us recall +the definition of Reed–Solomon codes that AG codes generalise. +3.1. Reed–Solomon codes. — +Definition 35. — Let α1, . . . , αn be distinct elements of Fq. Let 0 ⩽ k ⩽ n, the code RSk is defined as +RSk(α1, . . . , αn) +def += {(p(α1), . . . , p(αn)) | p ∈ Fq[x]⩽k−1}. + +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). However, they are constrained in the sense that the αi’s should be distinct and hence the +length should be bounded by q. 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. +Here, curves enter the game. Note first that Reed–Solomon codes may be defined in a much more pedant +manner as follows. Consider the projective line P1 and let P1, . . . , Pn be the rational points of P1 with +respective homogeneous coordinates (α1 : 1), . . . , (αn : 1). Then, RSk(α1, . . . , αn) may be defined as +RSk(α1, . . . , αn) = {(h(P1), . . . , h(Pn)) | h ∈ L((k − 1)P∞)} . +This leads to a natural generalisation to algebraic curves. 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. +3.2. Algebraic geometry codes. — We give a minimal introduction to algebraic geometry (AG) +codes. The reader interested in further references is encouraged to have a look at the surveys [HvLP98, +Duu08, CR21] or the books [TVN07, Sti09]. We also refer to [HP95, BH08] for references on the +decoding of AG codes. +Definition 36. — Let X be a smooth absolutely irreducible curve over Fq. Let P = (P1, . . . , Pn) be +an ordered sequence of distinct rational points of X . Let G be a rational divisor on X whose support +avoids the points P1, . . . , Pn. Then, the algebraic geometry code CL(X , P, G) is defined as +CL(X , P, G) +def += {(f(P1), . . . , f(Pn)) | f ∈ L(G)}. +Once the codes are defined, their parameters can be evaluated using the previously introduced material +of algebraic geometry. +Theorem 37. — Let X be a smooth absolutely irreducible curve of genus g over Fq. let P = (P1, . . . , Pn) +be a tuple of rational points of X and G be a rational divisor on X whose support avoids P1, . . . , Pn. +Suppose that deg G < n. 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; +(6) +d +⩾ +n − deg G. +(7) +Proof. — Denote by DP the divisor DP +def += P1 + · · · + Pn. Consider the map +evP : +� L(G) +−→ +Fn +q +f +�−→ +(f(P1), . . . , f(Pn)). +Its image is trivially CL(X , P, G). The kernel of this map is the subspace of L(G) of functions f vanishing +at P1, . . . , Pn. This subspace is nothing but L(G − DP). By assumption, deg(G − DP) = −(n − deg G), +is negative and hence, from Proposition 24 (ii), L(G − DP) = ker evP = {0}. Thus, evP is injective and +dim CL(X , P, G) = dim L(G) ⩾ deg G + 1 − g, +with equality if deg G > 2g − 2. Here, the last inequality together with the equality case are due to +Riemann–Roch Theorem (Theorem 30). +For the minimum distance, let us introduce h ∈ L(G)\{0} such that evP(h) has Hamming weight d. It +means that there exist distinct points Pi1, . . . , Pin−d among P1, . . . , Pn at which h vanishes. 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. +Let us comment this last result. It was mentioned in § 1.2 that, from Singleton bound (1), any [n, k, d]q +code satisfies +k + d ⩽ n + 1. + +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. +In summary, AG codes are in the worst case at “distance g from Singleton bound”. Thus, one can expect +good codes for a “not too large” genus g. 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. Thus, for the length to be large, we look for curves with the largest possible number of rational +points. +3.3. The problem of infinite sequence of curves with many points compared to their genus. +— We expect to get sequences of curves over Fq whose genus grows slowly and number of rational points +grows quickly. However, these two objectives are somehow in opposition: to get many rational points, +we need a large genus. 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. +Thus, we look for a good trade off between the genus and the number of rational points. 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. For this, let +us consider a sequence of curves (Xs)s∈N with sequence of genera (gs)s∈N. 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. 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, . . . , Pns) will be chosen as the full list of rational points, i.e. ns = ♯Xs(Fq). +Remark 38. — One could ask whether it is possible to have a rational divisor Gs of any degree whose +support avoids P1, . . . , Pns while {P1, . . . , Pns} = X (Fq)? 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. See [CR21, +Rem. 15.3.8] for a detailed discussion on this specific question. +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. +Therefore, one can eliminate deg Gs and get +(10) +ks + ds ⩾ ns + 1 − gs. +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). Therefore, any pair (δ, R) lying on the line of equation R + δ = 1 − 1 +γ is +achievable. +Remark 39. — 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. +Exercise 40. — 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 +3.4. The Ihara constant A(q). — Now, we would like to estimate the optimal asymptotic parameters +(δ, R) that can be achieved. 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). The remainder of these notes consists in outlining a proof +of the following statement. +Theorem 41. — For q = p2 and p a prime number, we have +A(q) ⩾ √q − 1. +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. +Let us conclude with some comments. +• Actually, the result extends to q = p2m for any m ⩾ 1 but the proof gets more complicated and +involves other families of curves. Namely, the proof to follow involves modular curves, while the +general case involves Shimura curves. See [Iha81, TVZ82]. +• The TVZ bound is actually optimal. 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. +• Another proof of Theorem 41 using a very different approach has been given by Garcia and +Stichtenoth in [GS95]. +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. +4. Elliptic curves +Elliptic curves is another fascinating topic in number theory. They are also a fundamental object +in cryptography but this is not the point of these notes. In this section, we start by presenting basic +notions about these objects over an arbitrary field. Our objective is in particular to construct these +so–called modular curves which will yield excellent codes. These modular curves are algebraic curves +which parameterise families of elliptic curves with a specific extra structure called level. +Afterwards, in Section 5, we will discuss elliptic curves and modular curves over C. This choice of +discussing complex curves in such notes might seem surprising while our interest will clearly be curves +over finite fields. However, a preliminary study of the complex case presents several advantages. First, it +provides a much more intuitive presentation of the topics with the benefits of the possible use of analytical +tools. Second, even if the analytic proofs cannot transpose in the finite field setting, they permit to +compute algebraic formulas, i.e. polynomial equations defining modular curves. 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. +Note. In this section, we assume the ground field K to have characteristic different from 2 and 3. Most +of the material of the present section and the subsequent one are taken from the book [Sil09] and the +lecture notes [Mil17]. +4.1. Basic definitions. — 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 . From Proposition 21, the Riemann–Roch space L(0) +associated to the zero divisor contains only the constant functions and hence has dimension 1. 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). Denote by x, y two functions such that +L(2OE ) = SpanK{1, x} +L(3OE ) = SpanK{1, x, y}. +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. +Now, consider the space L(6OE ). It contains the functions +1, x, y, x2, xy, x3, y2. +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. +Exercise 42. — 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. Deduce from this that a ̸= 0. +Now, we perform successive changes of variables which are admissible, i.e. changes of variables of the +form (11). 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. A change y ← y + v1 +2 yields +(13) +y2 = a2x3 + b2x2 + c2x + d2. +for some a2, b2, c2, d2 ∈ K. 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. 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. Such an equation is called a Weierstrass equation of the curve. +Exercise 43. — Using Exercise 42, check that the last change of variables was admissible, i.e. that +a3 ̸= 0. +In summary, starting from an elliptic curve E over K, i.e. 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 . These +functions are related by the relation (15) and hence the function y2 − x3 − Ax − B vanishes everywhere +on E . This leads to the following statement. +Theorem 44. — Let E be an elliptic over a field K of characteristic different from 2 and 3, i.e. 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. +(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. 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. + +CODES AND MODULAR CURVES +17 +Proof. — The fact that the image of E is contained in such a curve is a consequence of the previous +discussion. 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. III.3.1]. +Remark 45. — Geometrically speaking, the sequence of changes of variables can be interpreted as +follow. +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). 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 . +Remark 46. — In the sequel, we will not only consider Weierstrass form. 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. +4.2. The j–invariant. — In what follows, it will be important to classify elliptic curves up to isomor- +phism. For this sake, we introduce a fundamental invariant: the j–invariant. Reconsider a Weierstrass +equation (15) +y2 = x3 + Ax + B. +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. This leads to another Weierstrass equation y2 = +x3 + A′x + B′ where A′ = A +u4 and B′ = B +u6 . 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. III.1.4(a)(i)]). Hence, j is well–defined for any +elliptic curve since, by definition, such curves are smooth. 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. +We conclude this subsection by the following statements asserting that the j–invariant characterises an +elliptic curve over K in a unique manner. The proof is omitted and can be found in [Sil09, Prop. III.1.4(b- +c)]. +Proposition 47. — Two elliptic curves are isomorphic over K if and only if they have the same j– +invariant. Conversely, given j0 ∈ K, there exists an elliptic curve E over K(j0) with j–invariant j0. +Remark 48. — Note that two elliptic curves defined over K may be isomorphic over K without being +isomorphic over K. For instance, suppose that −1 is not a square in K. 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. Such curves are said to be a twist of each other. +Remark 49. — 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. + +18 +ALAIN COUVREUR +Figure 3. The addition law on an elliptic curve (Source: Cornelius Schätz blog) +4.3. The group law. — A remarkable feature of such curves is that they naturally have a group +structure. Namely, given an elliptic curve E over K, the set E (K) has a structure of abelian group. More +generally, for any algebraic extension L of K, then E (L) has an abelian group structure too. This group +structure is usually represented with a so–called chord and tangent process as represented by Figure 3. +It can be described as follows: +• Given two points P, Q ∈ E (L), draw the line L ⊆ A2 (or P2) joining them. If P = Q let L be the +tangent line of E at P. +• 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 . +• Let R be the point with coordinates (xR0, −yR0) if R0 ̸= OE and the point OE otherwise. This +point is defined to be the sum of P and Q. +Exercise 50. — (1) Prove that the intersection of E with a line is made of 3 points of P2 possibly +counted with multiplicity; +(2) Prove that R0 ∈ E (L); +This description is simple to understand but it is not completely obvious to prove that it provides a +group structure. In particular, the associativity is far from being obvious using this description. 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. Indeed, denote by DivK(E ) the group of rational divisors on E and by +Div0 +K(E ) the subgroup of divisors of degree 0. Finally denote by PrincK(E ) the group of principal divisors, +i.e. of divisors of the form (f) where f ∈ K(E ) \ {0}. 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 ). +Proposition 51. — Let E be an elliptic curve over K. Any element of Pic0 +K(E ) has a representative of +the form P − OE where P is some rational point in E (K). +Proof. — Let G ∈ Div0 +K(E ). +The divisor G + OE has degree 1 and, from Riemann–Roch Theorem +L(G + OE ) has dimension 1. Thus, there exists f ∈ L(G + OE ) \ {0}. By definition of L(G + OE ), the +function f satisfies +(f) + G + OE ⩾ 0. +The latter divisor is positive with degree 1 and hence equals some rational point P. Thus (f)+G = P−OE , +which entails that G and P − OE have the same class in Pic0 +K(E ). +(8)Possibly R0 equals P or Q. This is the reason why we mentioned 3 points counted with multiplicity. If the intersection +multiplicity of L with E at P (resp. Q) is 2 then, we set R0 +def += P (resp. Q). + +R +R=P+QCODES AND MODULAR CURVES +19 +Theorem 52. — Let P, Q ∈ E (K) and R be the sum of P + Q according to the previously introduced +addition law. Then, the classes of R − OE and (P − OE ) + (Q − OE ) are the same in Pic0 +K(E). +Proof. — From Exercise 50 (1), there is a point R0 which is contained in the line L joining P and Q. +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 ′. 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. 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. +As a conclusion, we have the following bijection: +� +E (K) +−→ +Pic0 +K(E ) +P +�−→ +P − OE +mod PrincK(E ) . +Via this bijection, we can equip E (K) with a group structure whose law is nothing but the previously +described chord–tangent one. Therefore, E (K) equipped with the chord–tangent law has a group structure +which is isomorphic to Pic0 +K(E ). +Remark 53. — 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. In particular, the whole set of geometric points E (K) has a structure of abelian +group. +4.4. Torsion and isogenies. — 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. For this sake, we first need +to discuss some specific subgroups of points of elliptic curves: their torsion subgroups. +4.4.1. Torsion subgroups. — 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). Interestingly, this group has a natural structure of +Z/ℓZ–module, and, in particular, is an Fℓ–vector space when ℓ is prime. The next theorem asserts that +this space has always dimension 2 when ℓ is prime to the characteristic. The proof of the next statement +is omitted. +Theorem 54. — Let E be an elliptic curve over K. Let ℓ be an integer. If ℓ is prime to the characteristic +of K, then +E [ℓ] ≃ Z/ℓZ × Z/ℓZ. +Else, if p denotes the characteristic of K and p ̸= 0, then +E [p] ≃ +� either +Z/pZ +or +0. +In the former case the curve is said to be ordinary, in the latter it is said to be supersingular. +4.4.2. Isogenies. — Given two elliptic curves E , E ′, an isogeny φ : E → E ′ is a morphism between these +curves sending the neutral element OE onto OE ′. Such a map is always surjective from E (K) into E ′(K). +A remarkable property is that such a map is necessarily a morphism of groups (see [Sil09, Thm. III.4.8]. +As any morphism of curves, an isogeny φ : E → E ′ induces a field extension K(E ′)/K(E ). 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. An isogeny of degree ℓ will usually be referred to as an ℓ–isogeny. +Example 55. — Taken from [Sil09, Ex. III.4.5]. Let a, b ∈ K, b ̸= 0 and a2 − 4b ̸= 0. Consider the +curves with equations: +E : y2 += +x3 + ax2 + bx +E ′ : y2 += +x3 − 2ax2 + (a2 − 4b)x. + +20 +ALAIN COUVREUR +The following map is a 2–isogeny: +(16) +� +E +−→ +E ′ +(x, y) +�−→ +� +y2 +x2 , y(b−x2) +x2 +� +. +Exercise 56. — Check that the map (16) actually sends E into E ′. Hint. Using a computer algebra +software may be helpful for this exercise. +Example 57. — 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). +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. If E is defined over Fq (i.e. if A, B ∈ Fq) then the +Frobenius map is an endomorphism of E . +Example 58. — 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. Its kernel is E [m]. +A separable isogeny of degree ℓ, regarded as a group morphism E (K) → E (K) is surjective with a +finite kernel of cardinality ℓ. Its kernel is a subgroup of E [ℓ]. +Theorem 59 ([Sil09, Prop. III.4.12]). — 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. The curve E ′ is sometimes +denoted as E /K. +Remark 60. — In the previous statement, further precision can be given on the field of definition of +E ′ and φ. 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). The field of definition of φ and E ′ is that of K. +Note that the field of definition is not the smallest field of definition of any geometric point of K. For +instance, there may be non rational m–torsion points while E [m] is defined over K. +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. +The existence and uniqueness of this map are proven in [Sil09, § III.6]. +Example 61. — In the case of a separable isogeny φ : E → E ′ of degree m, its kernel is a group with +m elements. By Lagrange Theorem, such a finite group is of m–torsion and hence ker φ ⊆ E [m]. 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. In particular E ′/φ(E [m]) ≃ E /E [m] ≃ E . The last isomorphism is +induced by the map P �→ mP. +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. In this context, these notions will be +much easier to visualise. +4.5. Elliptic curves over the complex numbers. — As already explained earlier, complex elliptic +curves is not the topic of this lecture. It is however necessary to discuss a bit about them. In order not +to spend too much time on the topic, many proofs of non trivial statements are omitted and replaced by +precise references. Clearly, the summary to follow is strictly included in Chapter VI of Silverman’s book +[Sil09]. + +CODES AND MODULAR CURVES +21 +4.5.1. Lattices and the Weierstrass ℘ function. — In the complex setting, an elliptic curve is isomorphic +to a complex torus. 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. 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.e. for any z ∈ C \ Λ and ω ∈ Λ, +℘(z + ω) = ℘(z). The proof of convergence of the series is left to the reader. +Note that, since ℘ is Λ–periodic, it passes to the quotient and induces a meromorphic function on +the torus C/Λ. 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. +Theorem 62. — 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. +Proof. — The series +℘(z) − 1 +z2 = +� +ω∈Λ\{0} +� +1 +(z − ω)2 − 1 +ω2 +� +is even and vanishes at 0. Hence, in the neighbourhood of 0, its Taylor series expansion depends only on +z2. 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). +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). +The function ℘′ +Λ(z)2 − 4℘Λ(z)3 − g2℘Λ(z) is Λ–periodic, meromorphic on C with pole locus contained +in Λ. From (17), it has no pole at 0 and, by Λ–periodicity has no pole at all and hence is holomorphic +on C. Since it continuous and Λ–periodic on C, it is bounded, and by Liouville’s theorem, it should be +constant. Therefore, there exists g3 ∈ C such that ℘′ +Λ(z)2 = 4℘Λ(z)3 + g2℘Λ(z) + g3. +Exercise 63. — Prove that a Λ–periodic holomorphic function is bounded on C. +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. +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). +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). 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. +Theorem 64. — The map ΨΛ defined in (18) is a biholomorphic isomorphism between C/Λ and the +elliptic curve E of equation y2 = 4x3+g2x+g3. 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 § 4.3. 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. +Proof. — See [Sil09, Prop. VI.3.6] for the group isomorphism. For the construction of a lattice from an +elliptic curve, see [Sil09, § VI.1]. +4.5.2. Torsion, isogenies. — 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. +Let us start with the torsion. From Theorem 54, for m prime to the characteristic, the m–torsion of +an elliptic curve is isomorphic to Z/mZ × Z/mZ. In the complex setting, consider a torus C/Λ. 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. Then, the torsion subgroup of C/Λ is isomorphic to +1 +mΛ/Λ. 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. +Now consider isogenies. When considering complex tori, isogenies are holomorphic maps C/Λ → C/Λ′. +It turns out that such maps lift to C and have a very particular structure. +Theorem 65. — Let Λ, Λ′ ⊂ C be two lattices and f : C/Λ → C/Λ′ be a holomorphic map 0 sending +onto 0. Then f lifts to a holomorphic map f0 : C → C such that +∀z ∈ C, +f0(z) +mod Λ′ = f(z +mod Λ). +Moreover, f0 is a similitude, i.e. there exists a ∈ C such that +∀z ∈ C, f0(z) = az. +Proof. — See [Sil09, Thm. VI.4.1]. +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′. +Example 66. — 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′. +From a similitude z �→ az such that aΛ ⊂ Λ′, the degree of the corresponding isogeny is given by +♯(Λ′/aΛ). In the previous example, the isogeny has degree 2. +Finally, let ℓ be a prime integer, and suppose that we have a degree–ℓ isogeny φ1 : C/Λ → C/Λ′. This +entails the existence of a ∈ C such that ♯(Λ′/aΛ) = ℓ. 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. +Exercise 67. — Prove the last assertion. +Now consider the map z �→ ℓ +az. It induces an isogeny φ2 : C/Λ′ → C/Λ of degree ℓ. 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. Therefore, +φ2 is nothing but the dual isogeny map ˆφ1 of φ1. + +CODES AND MODULAR CURVES +23 +4.6. Automorphisms. — Now, we have a nice description of morphisms of complex elliptic curves. +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. +One will also be interested in the sequel by automorphisms of an elliptic curve, which correspond to +similitudes z �→ az such that aΛ = Λ. One can prove that such an a satisfies |a| = 1. +Remark 68. — 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. In addition, if there exists +a ∈ C\Z such that aΛ ⊂ Λ, then the corresponding elliptic curve is said to be with complex multiplication. +An elementary automorphism for any complex torus is z �→ −z. 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). +Furthermore, some sporadic elliptic curves have nontrivial automophisms coming from z �→ az with +|a| = 1 and a /∈ {±1}. +Theorem 69. — Let C/Λ be a complex torus with an automorphism z �→ az with |a| = 1 and a /∈ {±1}. +Equivalently, the lattice Λ satisfies Λ = aΛ. Then, Λ is the image by a similitude of one of these two +lattices: +Z ⊕ Zi +or +Z ⊕ Zρ, +where ρ = e +iπ +3 . +Proof. — Let Λ be a lattice such that aΛ = Λ and ν ∈ Λ \ {0} be a vector of minimal modulus. Since we +look for Λ up to a similitude, one can assume that ν = 1 and that for all ω ∈ Λ \ {0}, |ω| ⩾ 1. Assuming +that 1 ∈ Λ, then, by assumption on Λ, we deduce that a, a2 are elements of Λ too. 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. Consequently, +a2 + 1 = −2Re(a)a. +Note that |a| = 1 and a /∈ R entails −1 < Re(a) < 1. If 2Re(a) /∈ Z, then there is ε ∈ {−1, 0, 1} such that +a2 + εa + 1 = γa +for some 0 < γ < 1. 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. Therefore Re(a) ∈ {− 1 +2, 0, 1 +2}. +Case Re(a) = 0 provides the case Λ = Z ⊕ Zi and the two other cases provide the same lattice, namely +Z ⊕ Zρ. +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). +The corresponding automorphisms being respectively +(x, y) +�−→ +(−x, iy) +(x, y) +�−→ +(ρx, −y). +Note that these automorphisms have respective orders 4 and 6 which are the multiplicative orders of i and +ρ. Finally, note that for any field containing fourth and sixth roots of 1, the curves with equations (19) +have a nontrivial automorphism group. Moreover, one can prove that they are the only curves with non +trivial automorphism groups [Sil09, Thm. III.10.1] and that their automorphism groups have respective +cardinalities 4 and 6. + +24 +ALAIN COUVREUR +5. Modular curves +5.1. The Poincaré upper half plane. — The objective is to classify elliptic curves over C up to +isomorphism. As explained in § 4.5, this reduces to classify lattices up to similitudes whose definition is +recalled there. +Definition 70 (Similitudes of C). — A similitude of C is a map of the form z �→ az for some a ∈ C×. +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. This requires to introduce another group action on the possible bases. 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). +Up to swapping the entries of the basis, one can always assume that the bases we consider have the +same orientation, i.e. that Im( ω1 +ω2 ) > 0 (resp. Im( µ1 +µ2 ) > 0), where Im(·) denotes the imaginary part of +a complex number. If the bases are chosen under this constraint, then the transition matrix M always +has a positive determinant and hence is in SL2(Z). 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). +Next, we need to consider the action of similitudes. 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. Let +H +def += {z ∈ C | Im(z) > 0} , +be the Poincaré upper half plane. 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. Starting from +M = +�a +b +c +d +� +∈ SL2(Z), +M acts on bases as: +M · +�ω1 +ω2 +� += +�aω1 + bω2 +cω1 + dω2 +� +. +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 § 4.5, 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. +5.2. The curve X0(1). — So, to parameterise the set of elliptic curves up to isomorphism, we can +consider the quotient SL2(Z)\H. It is proved in [Mil17, Prop. 2.21] that this quotient is a complex +variety isomorphic to A1, i.e. to the complex affine line. 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 . + +CODES AND MODULAR CURVES +25 +Figure 4. 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. In the complex setting, this is nothing +but a compactification and the affine line can be compactified with one point. 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. +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. Namely, we define +H∗ def += H ∪ P1(Q). +Next let us see how the action of SL2(Z) extends to P1(Q). +Proposition 71. — 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). +This action is transitive, i.e. for any (u : v), (u′ : v′) ∈ P1(Q), there exists M ∈ SL2(Z) such that +M · (u : v) = (u′ : v′). +Proof. — First, let us prove that the orbit of (0 : 1) equals the whole P1(Q). Note first that +�0 +−1 +1 +0 +� +· (0 : 1) = (−1 : 0) = (1 : 0). +Hence (1 : 0) is in the orbit of (0 : 1). Next, consider any other point (s : t) ∈ P1(Q) \ {(1 : 0)}, i.e. such +that t ̸= 0. 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. 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). +Therefore, any element of P1(Q) is in the orbit of (0 : 1). +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). +Therefore, the quotient SL2(Z)\H∗ is nothing but the compactification of SL2(Z)\H by adjoining a +single point. This quotient is usually denoted as X0(1) and is nothing but the Riemann sphere P1(C). +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. This map realises an +isomorphism between X0(1) and P1(C). It can be “made explicit” as follows. From τ ∈ H construct the +lattice Λτ = Z ⊕ Zτ. Then using the Weierstrass ℘Λτ function, compute an equation of the elliptic curve +corresponding to C/Λτ. Then, j(τ) is nothing but the j–invariant of this latter elliptic curve. + +-2 +-1 +0 +1 +226 +ALAIN COUVREUR +5.3. The curve X0(ℓ). — 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. 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. 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). +Remark 72. — 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. +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. Then, in the complex setting, it reduces to classify pairs of lattices Λ, Λ′ such that Λ ⊆ Λ′ +and ♯(Λ′/Λ) = ℓ. 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 . 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. 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. The elements of SL2(Fℓ) that fix the class of ω1 +ℓ are the upper triangular matrices. 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 +� +. +Namely, this is the group of elements of SL2(Z) which induce an automorphism of E [ℓ] fixing C. +Exercise 73. — Prove that the canonical map +SL2(Z) −→ SL2(Fℓ) +given by the reduction of the coefficients modulo ℓ is surjective. 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. +(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. +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. +This variety has a compactification +X0(ℓ) +def += Γ0(ℓ)\H∗. +This is a compact Riemann surface and it can be proved that such an object is actually algebraic, i.e. +is biholomorphic with a smooth complex projective curve. This structure of algebraic curve is discussed +further. + +CODES AND MODULAR CURVES +27 +The next statement gives a crucial information, namely the genus of X0(ℓ). +Theorem 74. — 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]. +We first need two technical lemmas. +Lemma 75. — Let ℓ be a prime integer. Let Λ ⊆ C be a lattice and Λ1, Λ2 be two distinct lattices both +containing Λ and ♯(Λ1/Λ) = ♯(Λ2/Λ) = ℓ. Suppose that aΛ1 = Λ2 for some a ∈ C. Then |a| = 1 and +aΛ = Λ. Equivalently, given an elliptic curve E over C and two distinct subgroups C1, C2 of cardinality +ℓ of E [ℓ]. If the curves E /C1 and E /C2 are isomorphic, then there is an automorphism of E sending C1 +onto C2. +Remark 76. — Note that if E has such an automorphism, then it should be one of the two curves +mentioned in Theorem 69. +Proof of Lemma 75. — Step 1. An adapted basis. 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. 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. +Next, we claim that +Λ2 = Zη1 ⊕ Zuη1 + η2 +ℓ +, +for some u ∈ {0, . . . , ℓ − 1}. Indeed, consider +� 1 +ℓ Λ +� +/Λ, which isomorphic to Fℓ × Fℓ. In this quotient, +Λ1/Λ and Λ2/Λ are identified to two Fℓ–subspaces of dimension 1 in direct sum. 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. This implies that there exists +u ∈ {0, . . . , ℓ − 1} such that uη1+η2 +ℓ +∈ Λ2 and hence +Zη1 ⊕ Zuη1 + η2 +ℓ +⊆ Λ2. +Then, since ♯(Λ2/Λ) = ℓ, we can deduce that the above inclusion is actually an equality. Finally, define +ω1, ω2 as +�ω1 +ω2 +� +def += +�1 +u +0 +1 +� +· +�η1 +η2 +� +. +Note that the above change of variables is given by a matrix in SL2(Z) and provides a basis for Λ which +satisfies (21). +Step 2. The modulus of a. A classical notion in lattice theory is that of the determinant or volume +of the lattice. It can be defined as follows. Consider Λ = Zω1 ⊕ Zω2 and regard C as a 2–dimensional +R–vector space with canonical basis (1, i). Since any basis of Λ can be deduced from (ω1, ω2) by applying +a matrix in GL2(Z), i.e. a matrix with determinant ±1, the quantity | det(ω1, ω2)| is the same for any +basis of Λ. Hence we denote this quantity | det Λ|. From (21), we have +(22) +det Λ1 = 1 +ℓ det Λ = det Λ2. +Moreover, the multiplication by a map z �→ az regarded as an R–linear endomorphism of C has determi- +nant |a|2. 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. Next, the assumption Λ2 = aΛ1 together with (22) give +det Λ1 = det Λ2 = |a|2 det Λ1, +which yields |a|2 = 1. +Step 3. We aim to prove that aΛ = Λ. Suppose it does not. Since Λ ⊆ Λ1, Λ ⊆ Λ2 and aΛ ⊆ aΛ1 = Λ2, +then Λ + aΛ ⊆ Λ2. Recall that ♯Λ2/Λ = ℓ and ℓ is prime. Then, since we assumed that Λ ⊊ Λ + aΛ, we +get Λ+aΛ = Λ2. Similarly, one deduces that a−1Λ+Λ = Λ1. Next, from (21), we see that Λ1 +Λ2 = 1 +ℓ Λ +and hence +a−1Λ + Λ + aΛ = 1 +ℓ Λ. +By induction, +a−sΛ + · · · + a−1Λ + Λ + aΛ + · · · + asΛ = 1 +ℓs Λ. +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. +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. When N goes to infinity, the right hand side is a convergent series. Thus, so +does the left hand side and hence, its general term should go to 0. 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. Therefore, +the sequence of partial sums of the left–hand side is stationary while that of the right–hand side is note. +This is a contradiction. Therefore aΛ = Λ. +Lemma 77. — Let Ei +def += C/(Z ⊕ Zi) and consider its automorphism group Gi induced by the multipli- +cations by {±1, ±i} in C. Then, any P ∈ Ei(C) which has a non trivial stabiliser under the action of Gi +is in E [2]. +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]. +Proof. — In the case Ei, denote by Λi +def += Z ⊕ Zi. +Since Gi is cyclic of order 4, its only nontrivial +subgroups are {±1} and Gi itself. A point P ∈ Ei(C) stabilised by {±1} corresponds to z ∈ C such that +z ≡ −z mod Λi. That is to say 2z ∈ Λi and hence P ∈ Ei[2]. Similarly if P is stabilised by all Gi it is a +fortiori stabilised by {±1} and hence should be in Ei[2]. +Consider now the case of Eρ. Denote by Λρ +def += Z ⊕ Zρ. Since Gρ is cyclic of order 6, its only possible +nontrivial subgroups are {±1}, {1, ρ2, ρ4} and Gρ itself. Let us consider points which are stabilised by +one of these groups. +Let P ∈ Eρ(C) stabilised by {±1}, then the very same reasoning as for Ei yields P ∈ Eρ[2]. +Let P ∈ Eρ(C) stabilised by {1, ρ2, ρ4}. +This corresponds to z ∈ C satisfying ρ2z ≡ z mod Λρ. +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. +This entails that +� +−b += +a + µ +a − b += +b + ν, + +CODES AND MODULAR CURVES +29 +where µ, ν ∈ Z. By elimination, we deduce that 3a ∈ Z and 3b ∈ Z, that is to say z ∈ 1 +3Λρ and hence +P ∈ Eρ[3]. +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. +Proof of Theorem 74. — 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 . This map is algebraic (this +will appear more naturally in § 5.4). The objective is to apply Riemann Hurwitz formula (Theorem 31) +to π in order to compute the genus of X0(ℓ). +Step 1. The degree of π. The degree of π is the generic number of pre-images of a point of X0(1). 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]. From Lemma 75, if E has no nontrivial automorphism, then two distinct subgroups +C1, C2 provide non isomorphic pairs (E , C1), (E , C2). 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]. Since ℓ is prime, then from +Theorem 54, E [ℓ] ≃ Fℓ × Fℓ and hence is a vector space of dimension 2 over Fℓ. 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.e. of lines) of Fℓ × Fℓ equals ♯P1(Fℓ) = ℓ + 1. Thus, +deg π = ℓ + 1. +Now, the map is ramified at the points corresponding to curves with nontrivial automorphisms and +possibly the point at inifinity. From Theorem 69, the curves with nontrivial automorphisms correspond +to the tori C/(Z ⊕ Zi) and C/(Z ⊕ Zρ), where ρ = e +iπ +3 . For these tori, we need to understand the action +of the automorphisms on the ℓ–torsion. +Step 2. Ramification at C/(Z ⊕ Zi). The curve is equipped with a nontrivial automorphism η of +order 4, which corresponds to the multiplication by i in C. This automorphism acts on the ℓ–torsion +and, from [Sil09, Thm. III.4.8], such an automorphism is a group automorphism and hence its action +on E [ℓ] regarded as an Fℓ–vector space is Fℓ–linear. We denote by ηℓ, the automorphism η restricted to +E [ℓ]. From Lemma 77, since ℓ > 3 any point in E [ℓ] \ {OE } has has an orbit of cardinality 4 under ηℓ. +Therefore, ηℓ has order 4 and two situations may occur. 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. 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. Two lines of E [ℓ] in a same orbit under ηℓ correspond to a +same point in X0(ℓ). This point is a ramification point with ramification index 2. 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. +Otherwise ℓ ≡ 3 mod 4. In this situation ηℓ has no eigenspace in E [ℓ] and the orbit of any line has +cardinality 2. Thus, there are ℓ+1 +2 +points above E which are all ramified with ramification index 2. +Step 3. Ramification at C/(Z ⊕ Zρ). Here we have an automorphism η of order 6 and denote again +by ηℓ its restriction to E [ℓ]. Here again, from Lemma 77, we know that ηℓ has also order 6. In this +situation, if ℓ ≡ 1 mod 3, then Fℓ contains sixth roots of unity and ηℓ is diagonalisable. Therefore, the +two eigenspaces of ηℓ are left invariant and any other Fℓ–line of E [ℓ] has an orbit of cardinality 3. Indeed, +here again ρ3 = −Id and hence leaves any line globally invariant. 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. +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. + +30 +ALAIN COUVREUR +Step 4. Ramification at infinity. 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. We wish to estimate the number +of orbits in P1(Q) under the action of Γ0(ℓ). We claim that their number is 2, namely, the orbit of (0 : 1) +and that of (1 : 0). 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}. +One easily sees that the two orbits form a partition of P1(Q). 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. +Final step. Computation of the genus. Denote by gℓ the genus of X0(ℓ) and by g1 = 0 that of +X0(1). 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. 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. +An easy but cumbersome calculation treating separately the four cases ℓ ≡ 1, 5, 7, 11 mod 12 yields the +expected result. +5.4. The modular equation. — 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. We refer +the reader to [Mil17, Thm. 6.1] for a proof. +Theorem 78. — 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. +Remark 79. — A database of the polynomials Φℓ for small values of ℓ is available on Andrew Suther- +land’s webpage: https://math.mit.edu/∼drew/ClassicalModPolys.html +Let us give some comments about this statement. First, note that the projective closure of the complex +curve of equation Φℓ(x, y) = 0 is a “singular model” for X0(ℓ). Indeed, any point of X0(ℓ) corresponds to +an isomorphism class of ℓ–isogeny E → E ′. This yields a rational map +� +X0(ℓ) +−→ +P2(C) +(E → E ′) +�−→ +(j(E ) : j(E ′) : 1). +The image of this map is contained into the curve with equation Φℓ(x, y) = 0. However, this latter curve +is full of singularities and hence is not isomorphic to X0(ℓ). 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. +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. 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. 5.9]). 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. +These curves over Fp will be the objects of interest in order to prove the main theorem of this course, +namely Theorem 41. +Finally, the reader interested in a rigorous study of the reductions of modular curves cannot avoid the +language of schemes. For such a development, we refer the reader to the article of Celgene and Rapoport +[DR73] or the book of Katz and Mazur [KM85]. + +CODES AND MODULAR CURVES +31 +6. Proof of the main Theorem +Now, we almost have the material to prove Theorem 41. We have our family of curves X0(ℓ) for ℓ a +prime integer distinct from the characteristic p. +6.1. Genus of modular curves over finite fields. — Let us briefly discuss the genus of the curve. +The discussion to follow is far from being trivial. Thus, the reader is encouraged first to directly admit +the conclusion. +Namely that the genus of a modular curve over a finite field is that of its complex +counterpart. Let us briefly sketch the reasons why this holds. +It is known (see for instance in [Mor90, Thm. 5.9]) 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. +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. The genus introduced by Definition 25 in § 2.7 is the +geometric one. The arithmetic genus, which can be defined for instance from the Hilbert function of the +variety [Har77, Ch. IV], is always larger than or equal to the geometric one and they coincide if and +only if the curve is smooth. +Next, Grauert Theorem [Har77, Cor. III.12.9] permits to assert that the reduction modulo p of the +aforementioned model X0(ℓ) has the same arithmetic genus as the complex curve itself. Moreover, since +this model and its reduction are smooth, they also have the same geometric genus. 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. +6.2. The locus of supersingular curves. — There remains to get an estimate of the number of +rational points of such curves. For this we will focus on Fp2–points since their number can be bounded +from below using the two following statements. +Proposition 80. — Let E be a supersingular elliptic curve over Fp, then E is defined over Fp2. +Proof. — By definition, a supersingular curve E satisfies E [p] = {0}. Therefore, the multiplication by p +map [p] : E → E is totally inseparable. +Consider now the Frobenius map +φ : +� +E +−→ +E (p) +(x, y) +�−→ +(xp, yp). +It is a degree p isogeny, hence it has a dual isogeny ˆφ such that ˆφ◦φ = [p]. 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). +Thus, E (p2) = E and hence E is defined over Fp2. +Theorem 81. — 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. +Proof. — See [Sil09, Thm. V.4.1]. +6.3. Proof of the main theorem. — With these two last statements at hand we can finally provide +the proof of Theorem 41. We restrict the proof to the case p ⩾ 5. Note that Tsfasman–Vlăduţ–Zink +theorem remains true when p = 2, 3 but the coding theoretic interest is rather limited. +Proof of Theorem 41. — Consider the sequence of curves X0(ℓ) for ℓ ≡ 11 mod 12. From Theorem 74 +it has genus gℓ = ℓ+1 +12 . 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. 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. Depending on the class of p modulo 12 the curves with j–invariant 0 and +1728 may be supersingular. More precisely, from [Sil09, Ex. V.4.4 & V.4.5], +• for p ≡ 1 mod 12, both curves are ordinary (i.e. non supersingular) and then any supersingular +curve has no nontrivial automorphism and hence has ℓ + 1 pre-images in X0(ℓ)(Fp2). 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. +Therefore, there are p−5 +12 + 1 supersingular curves and all of them but one have ℓ + 1 distinct pre- +images. The remaining curve is the one with j = 0 and an automorphism group of order 6. Its +treatment is very similar to the proof of Theorem 74. Consider the action of the automorphism of +order 6 on the ℓ–torsion. From Lemma 77, this induces an automorphism η or order 6 of E [ℓ] ≃ F2 +ℓ. +Since ℓ ≡ 11 mod 12, then ℓ ≡ 2 mod 3 and hence Fℓ does not contains the sixth roots of unity. +Thus, η is not diagonalisable and hence cannot fix a line. Consequently, one proves that the orbit +of any line is the union of 3 distinct lines (η3 = −Id, which fixes the lines). 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. 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. Then +♯X0(ℓ)(Fp2) +gℓ +⩾ +p−1 +12 (ℓ + 1) +� ℓ+1 +12 +� += p − 1. +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. Which turns out to +be optimal. +References +[BH08] +Peter Beelen and Tom Høholdt. The decoding of algebraic geometry codes. 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