Selected Unsolved Problems in Coding Theory pp 145176  Cite as
Codes from Modular Curves
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Abstract
One of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the socalled modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraicgeometric codes (“AG codes”) constructed from modular curves can have parameters which beat the Gilbert–Varshamov lower bound if the ground field is sufficiently large.
Keywords
Congruence Subgroup Modular Curve Rational Compactification Shimura Variety Modular CurfOne of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the socalled modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraicgeometric codes (“AG codes”) constructed from modular curves can have parameters which beat the Gilbert–Varshamov lower bound if the ground field is sufficiently large.
Open Problem 30
Find an infinite family of binary linear codes which, asymptotically, beats the Gilbert–Varshamov lower bound or prove that no such family exists.
We shall try to make a more precise statement of this open problem below (see also Open Question 25 above). However, the basic idea is to try to use the theory of algebraic curves over a finite field for improvement of the Gilbert–Varshamov lower bound. Since we do not (yet) really know how to tell if an arbitrarily given code arises as an AG code [PSvW], perhaps all this sophistication can be avoided.
The topic of AG codes is sketched briefly in Huffman and Pless [HP1], Chap. 13. More complete treatments are given in Tsfasman, Vladut, and Nogin [TVN], Tsfasman and Vladut [TV], Stichtenoth [Sti], Moreno [Mo], and Nieddereiter and Xing [NX]. These are all recommended texts for background below.
6.1 An Overview
Modular curves are remarkable for many reasons, one of which is their high degree of symmetry. In other words, there are a large number of different automorphisms of the curve onto itself. When these curves are used to construct codes, those codes can display not only unusually good errorcorrecting ability but also remarkable symmetry properties. Still, there are aspects of the structure of this symmetry which are still unknown.
Let N>5 be a prime. The modular curve X(N) has a natural action by the finite group G=PSL(2,N), namely the projective special linear group with coefficients in GF(N). In fact, the quotient X(N)/G is isomorphic to X(1)≅ℙ^{1}. If D is a PSL(2,N)invariant divisor on X(N), then there is a natural representation of G on the Riemann–Roch space L(D). In this chapter, we discuss some results about the PSL(2,N)module structure of the Riemann–Roch space L(D) (in the case where N is prime and N≥7).
The Gmodule structure of L(D) is explicitly known if, in addition, \(N\equiv1\pmod{4}\). This is discussed further below.
In Sects. 6.2.1 and 6.4, applications to AG codes associated to this curve are considered (SAGE [S] was used to do some of these computations). In Sect. 6.7.1, we look at the examples N=7,11, using [GAP] to do many of the computations.
Notation
Throughout this chapter, we assume that N>5 is a prime, GF(N) is the finite field with N elements, and G=PSL(2,N).
6.2 Introduction to Algebraic Geometric Codes
Let \({\mathbb{F}}= \mathit{GF}(q)\) denote a finite field, and let \(F=\overline{{\mathbb{F}}}\) denote the algebraic closure of GF(q).
In the early 1980s, a Russian mathematician Goppa discovered a way to associated to each “nice” algebraic curve defined over a finite field a family of errorcorrecting codes whose length, dimension, and minimum distance can either be determined precisely or estimated in terms of some geometric parameters of the curve you started with. In this section, rather than going into detail about Goppa’s general construction, we shall focus on a very special case where these constructions can be made very explicitly.
6.2.1 The Codes
For properties of this map, see [JKT]. In particular, the following is known.
Lemma 177
IfDandEsatisfy\(\deg(D)>2g\)and\(\deg(E)>2g+2\), thenφand eval _{ E }are injective.
Proof
Let P be the permutation automorphism group of the code C=C(D,E) defined in (6.2.2). In many cases it is known that the map φ:G→P is an isomorphism (see, for example, [JKT]). In any case, using (6.2.4), we regard C as a Gmodule. In particular, the (bijective) evaluation map eval _{ E }:L(D)→C in (6.2.3) is Gequivariant.
A code associated to the projective line in the manner above is referred to as a generalized Reed–Solomon code. The automorphism groups of such codes have been completely determined (see, for example, [JKT]).
6.2.2 The Projective Line
By way of introduction, we start with the example of the simplest curve, the projective line.
What exactly is the projective line ℙ^{1}? The analogy to keep in mind is that ℙ^{1} is analogous to the complex plane compactified by adding the point at infinity, i.e., the Riemann sphere \(\hat{\mathbb{C}}\).
Algebraically, in a rigorous treatment points are replaced by places, “valuations” on the function field F(ℙ^{1}), which correspond to localizations of a coordinate ring F[ℙ^{1}] (see Moreno [Mo], Sect. 1.1).
We shall, for reasons of space, emphasize intuition over precision. What is a point? ℙ^{1} (as a set) may be thought of as the set of lines through the origin in affine space F^{2}. We say that two points in F^{2}−{(0,0)} are “equivalent” if they lie on the same line (this is an equivalence relation). If \(y\not=0\), then we denote the equivalence class of (x,y) by [a:1], where a=x/y. If y=0, then we denote the equivalence class of (x,y) by [1:0]. This notation is called the projective coordinate notation for elements of ℙ^{1}.
The group GL(2,ℂ) acts on the Riemann sphere by linear fractional (“Möbius”) transformations, \(z\longmapsto\frac{az+b}{cz+d}\), Open image in new window . This action factors through PGL(2,ℂ) since scalar matrices act trivially. Similarly, PGL(2,F) acts on ℙ^{1}. In fact, Aut(ℙ^{1})=PGL(2,F).
Riemann–Roch Spaces
A divisor on ℙ^{1} is simply a formal linear combination of points with integer coefficients, only finitely many of which are nonzero. The divisor off is the formal sum of zeros of f minus the poles, counted according to multiplicity. These sums include any zero or pole at the “point at infinity” on ℙ^{1}. For any given divisor D, the set of points occurring in the formal sum defining D whose integer coefficient is nonzero is called the support of D, written supp (D). The divisor of a rational function f is denoted div (f). If f is, for example, a polynomial of degree n in x, then div (f)=P_{1}+⋯+P_{ n }−n∞ and supp (div (f))={P_{1},…,P_{ n },∞}, where the P_{ i }’s denote the zeros of f. Since divisors are merely formal integral combinations of points, the sum and difference of any two divisors are other divisors. The Abelian group of all divisors is denoted Div (ℙ^{1}).
Let X=ℙ^{1}, and let F(X) denote the function field of X (the field of rational functions on X).
The Action of G on L(D)
Let F be algebraically closed, and let X=ℙ^{1}/F, by which we mean ℙ^{1} with base field F, so the field of rational functions is F(ℙ^{1}). In this case, Aut(X)=PGL(2,F).
Note that Y=X/G is also smooth and F(X)^{ G }=F(Y).
Lemma 178
 (a)IfDis effective, thenis a basis forL(D).$$\bigl\{1,m_{P_i}(x)^k \, \, 1\leq k\leq a_i,0\leq i\leq s\bigr\}$$
 (b)IfDis not effective but\(\deg(D)\geq0\), then writeD=dP+D′, where\(\deg(D')=0\), d>0, andPis any point. Letq(x)∈L(D′) (which is a onedimensional vector space) be any nonzero element. Thenis a basis forL(D).$$\bigl\{m_P(x)^{i}q(x)\,  \,0\leq i\leq d\bigr\}$$
 (c)
If\(\deg(D)<0\), thenL(D)={0}.
The first part is Lemma 2.4 in [Lo]. The other parts follow from the definitions and the Riemann–Roch theorem.
6.3 Introduction to Modular Curves
Suppose that V is a smooth projective variety over a finite field \({\mathbb{F}}\). An important problem in arithmetical algebraic geometry is the calculation of the number of \({\mathbb{F}}\)rational points of V, \(V({\mathbb{F}})\). The work of Goppa [G1] and others have shown its importance in geometric coding theory as well.^{1} We refer to this problem as the counting problem. In most cases it is very hard to find an explicit formula for the number of points of a variety over a finite field.
When the V arises by “reduction mod p” from a “Shimura variety” defined by certain group theoretical conditions (see Sect. 6.3.1 below), methods from nonAbelian harmonic analysis on groups can be used to find an explicit solution for the counting problem. The Arthur–Selberg trace formula [Shok] provides such a method. Using the Arthur–Selberg trace formula, an explicit formula for the counting problem has been found for Shimura varieties, thanks primarily to the work of Langlands and Kottwitz [Lan1, K1, K2].^{2} The trace formula allows one (with sufficient skill and expertise!) to relate, when V is a Shimura variety, the geometric numbers V(k) to orbital integrals from harmonic analysis ([Lab], for example), or to a linear combination of coefficients of automorphic forms ([Gel], for example), or even to representationtheoretic data ([Cas2], for example).
However, another type of application of the trace formula is very useful to the coding theorist. Moreno [Mo] first applied the trace formula in the context of Goppa codes to obtaining a new proof of a famous result of M. Tsfasman, S. Vladut, T. Zink, and Y. Ihara. (Actually, Moreno used a formula for the trace of the Hecke operators acting on the space of modular forms of weight 2, but this can be proven as a consequence of the Arthur–Selberg trace formula [DL], Sect. II.6.) This will be discussed below. We are going to restrict our attention in this chapter to the interplay between Goppa codes of modular curves, the counting problem, and the action of the automorphism group on these codes. We will give some examples using SAGE. In coding theory, curves with many rational points over finite fields are being used for construction of codes with some good specific characteristics. We discuss AG (or Goppa) codes arising from curves, first from an abstract general perspective, then turning to concrete examples associated to modular curves. We will try to explain these extremely technical ideas using a special case at the level of a typical graduate student with some background in modular forms, number theory, group theory, and algebraic geometry. For an approach similar in spirit, though from a more classical perspective, see the book of Moreno [Mo].
6.3.1 Shimura Curves
In this section we study arithmetic subgroups, arithmetical quotients, and their rational compactifications. Ihara first introduced Shimura curves, a rational compactification of Γ\ℍ where Γ is a particular discrete subgroup acting on the upper halfplane^{3} ℍ, from a classical perspective. We shall recall them from both the classical and grouptheoretical point of view. The latter perspective generalizes to higherdimensional Shimura varieties [Del].
Arithmetic Subgroups
In general, an arithmetic subgroup of SL(2,ℝ) is any discrete subgroup Γ that is commensurable with SL(2,ℤ), where commensurability means that the intersection Γ∩SL(2,ℤ) is of finite index in both Γ and SL(2,ℤ). The group Γ(N) has the property of being commensurable with SL(2,ℤ).
Riemann Surfaces as Algebraic Curves
We emphasize that the action of SL(2,ℝ) on ℍ is transitive, i.e., for any two points w_{1},w_{2}∈ℍ, there is an element g∈SL(2,ℝ) such that w_{2}=g⋅w_{1}. This can easily be proved. We also emphasize that there are subgroups of SL(2,ℝ) for which the action is not transitive; among them, the class of arithmetic subgroups are to be mentioned. For example, the group SL(2,ℤ) does not act transitively on ℍ, and the set of orbits of the action of SL(2,ℤ) on ℍ (and similarly any arithmetic subgroup) is infinite. We call the arithmetic quotientΓ\ℍ the set of orbits of the action of an arithmetic subgroup Γ on ℍ.
Example 179
We now recall the basic ideas that turn an arithmetic quotient of the form Γ\ℍ into an algebraic curve. Let Γ⊂SL(2,ℚ) be an arithmetic subgroup. The topological boundary of ℍ is ℝ and a point ∞. For the rational compactification of ℍ, we do not need to consider all the boundaries ℝ and {∞}. In fact, we need only to add to ℍ the cusps of Γ (a cusp of Γ is an element of ℚ that is fixed under the action of an element γ∈Γ with the property that tr (γ)=2). Any two cusps x_{1},x_{2} such that δ⋅x_{2}=x_{1} for an element δ∈Γ are called equivalent. Let C(Γ) be the set of inequivalent cusps of Γ. Then C(Γ) is finite. We add this set to ℍ and form the space ℍ^{∗}=ℍ∪C(Γ). This space will be equipped with certain topology such that a basis of the neighborhoods of the points of ℍ^{∗} is given by three types of open sets; if a point in ℍ^{∗} is lying in ℍ, then its neighborhoods consist of the usual open discs in ℍ; if the point is ∞, i.e., the cusp ∞, then its neighborhoods are the set of all points lying above the line Im (z)>α for any real number α; if the point is a cusp different from ∞ which is a rational number, then the system of neighborhoods of this point are the union of the cusp and the interior of a circle in ℍ tangent to the cusp. Under the topology whose system of open neighborhoods we just explained, ℍ^{∗} becomes a Hausdorff nonlocally compact space. The quotient space Γ\ℍ^{∗} with the quotient topology is a compact Hausdorff space. We refer to this compact quotient as the rational compactification of Γ\ℍ. For a detailed discussion, we refer the reader to [Shim].
When the arithmetic group is a congruence subgroup of SL(2,ℤ), the resulting algebraic curve is called a modular curve. For example, the rational compactification of Y(N)=Γ(N)\ℍ is denoted by X(N), and the compactification of Y_{0}(N)=Γ_{0}(N)\ℍ by X_{0}(N).
Example 180
Let N=1. Then Γ=Γ(1)=SL(2,ℤ). In this case, C(Γ)={∞}, since all rational cusps are equivalent to the cusp ∞. So ℍ^{∗}=ℍ∪{∞}, and Γ\ℍ^{∗} will be identified by Γ\ℍ∪{∞}. This may be seen as adding ∞ to the fundamental domain \(\mathcal{F}_{1} = \mathcal{F}\) of SL(2,ℤ) that consists of all complex numbers z∈ℍ in the upper halfplane with z≥1 and \(\mathrm{Re}\,(z) \leq\frac{1}{2}\).
The rational compactification of Γ\ℍ turns the space Γ\ℍ^{∗} into a compact Riemann surface (cf. [Shim]) and so into an algebraic curve (cf. [Nara] or [SS]).
In general, it is easiest to work with those arithmetic subgroups which are torsion free, and we shall assume from this point on that the arithmetic subgroups we deal with have this property. For example, Γ(N) and Γ_{0}(N) for N≥3 are torsion free.
An Adelic View of Arithmetic Quotients
Definition 181

What field are the curves X(N), X_{0}(N) “naturally” defined over?

How can they be described explicitly using algebraic equations?
Example 182
Models of genus 1 modular curves
Level  Discriminant  Weierstrass model  Reference 

11  −11  y^{2}+y=x^{3}−x^{2}  [BK], Table 1, p. 82 
14  −28  y^{2}+xy−y=x^{3}  p. 391, Table 12.1 of [Kn] 
15  15  y^{2}+7 xy+2 y=x^{3}+4 x^{2}+x  p. 65, Table 3.2 of [Kn] 
17  17  y^{2}+3 xy=x^{3}+x  p. 65, Table 3.2 of [Kn] 
19  −19  y^{2}+y=x^{3}+x^{2}+x  [BK], Table 1, p. 82 
20  80  y^{2}=x^{3}+x^{2}−x  p. 391, Table 12.1 of [Kn] 
21  −63  y^{2}+xy=x^{3}+x  p. 391, Table 12.1 of [Kn] 
24  −48  y^{2}=x^{3}−x^{2}+x  p. 391, Table 12.1 of [Kn] 
27  −27  y^{2}+y=x^{3}  p. 391, Table 12.1 of [Kn] 
32  64  y^{2}=x^{3}−x  p. 391, Table 12.1 of [Kn] 
36  Sect. 4.3 in [Ro]  
49  Sect. 4.3 in [Ro] 
For deeper study of Shimura varieties and the theory of canonical models, we refer the reader to [Del, Lan2], and [Shim].
6.3.2 Hecke Operators and Arithmetic on X_{0}(N)
In this section we recall some wellknown though relatively deep results on X_{0}(N)(GF(p)), where p is a prime not dividing N. These shall be used in the discussion of the Tsfasman, Vladut, Zink, and Ihara result later.
Theorem 183
(“Congruence relation” of Eichler–Shimura [Mo], Sect. 5.6.7, or [St1])
Example 184
One may try to compute the trace of the Hecke operators T_{ p } acting on the space of holomorphic cusp forms of weight 2, S_{2}(Γ_{0}(N)), by using either the Eichler–Shimura congruence relation, which we give below (see Theorem 183), or by using some easier but ad hoc ideas going back to Hecke which work in special cases. One simple idea is noting that S_{2}(Γ_{0}(N)) is spanned by simultaneous eigenforms of the Hecke operators (see, for example, Proposition 51 in Chap. III of [Kob]). In this case, it is known that the Fourier coefficient a_{ p }, p prime not dividing N, of a normalized (to have leading coefficient a_{1}=1) eigenform is the eigenvalue of T_{ p } (see, for example, Proposition 40 in Chap. III of [Kob]). If S_{2}(Γ_{0}(N)) is onedimensional, then any element in that space f(z) is such an eigenform.
The modular curve X_{0}(11) is of genus 1, so there is (up to a nonzero constant factor) only one holomorphic cusp form of weight 2 in S_{2}(Γ_{0}(11)) (see Theorem 186 below). There is a wellknown construction of this form (see [O2] or [Gel], Example 5.1), which we recall below. As we noted above, the pth coefficient a_{ p } (p a prime distinct from 11) of its Fourier expansion is known to satisfy a_{ p }=Tr(T_{ p }). These will be computed using SAGE.
For a representationtheoretic discussion of this example, see [Gel], Sect. 14.
For an example of an explicit element of S_{2}(Γ_{0}(32)), see Koblitz [Kob] (Sect. 5 in Chap. II and (3.40) in Chap. III). For a remarkable theorem which illustrates how far this ηfunction construction can be extended, see Morris’ theorem in Sect. 2.2 of [Ro].
To estimate \(a_{p^{k}}\), one may appeal to an explicit expression for \(\mathrm{Tr}(T_{p^{k}})\) known as the “Eichler–Selberg trace formula”, which we discuss next.
6.3.3 Eichler–Selberg Trace Formula
In this subsection, we recall the version of the trace formula for the Hecke operators due to Duflo and Labesse [DL], Sect. 6.

f(γz)=(cz+d)^{ k }f(z) for all Open image in new window , x∈H,

f is a holomorphic cusp form.
Theorem 185
(Eichler–Selberg trace formula)
Remark 24
Theorem 186
(Hurwitz–Zeuthen formula [Shim])^{7}
6.3.4 Modular Curves X(N)
Example 187
The curve X(N) over a field k parameterizes pairs of an elliptic curve over k and a subgroup of order N of the group structure on the elliptic curve. This can be extended to fields of positive characteristic if X(N) has good reduction. Since Klein’s equations have integer coefficients, they can also be extended to an arbitrary field k. However, Velu [V] (see also Ritzenthaler [R3]) has shown that X(N) has good reduction over fields of characteristic p where p does not divide N (in our case, \(p\not= N\), since N is itself assumed to be a prime).
Definition 188
When X has good reduction to a finite field k and, in addition, the characteristic ℓ of k does not divide G, we say that ℓ is good.
If k is a field of good characteristic, the automorphism group of X(N) is known to be PSL(2,N) [BCG].
Remark 25
When the formulas for the special cases Open image in new window , Open image in new window , and Open image in new window are written down separately, the similarity with the Weil representation for SL_{2}(ℤ/Nℤ) is striking (see also [A2]).
6.4 Application to Codes
In this section we discuss connections of our previous results with the theory of errorcorrecting codes.
Assume that ℓ is a good prime. Also, assume that k contains all the character values of G and that k is finite, where k denotes the field of definition of the reduction of X mod ℓ. (The point is that we want to be able to work over a separable algebraic closure \(\overline{k}\) of k but then be able to take \(\mathrm{Gal}(\overline{k}/k)\)fixed points to obtain our results.) We recall some background on AG codes following [JKT, JK1, JK2].
As an amusing application of our theory, we show how to easily recover some results of Tsfasman and Vladut on AG codes associated to modular curves.
In fact, if D is “sufficiently large” (so, D is nonspecial, and both φ and eval _{ E } are injective), then the Brauercharacter analogs of formulas in Joyner and Ksir [JK1] give not only the Gmodule structure of each L(rD_{∞}), but that of C as well.
See also Remark 4.1.66 in [TV].
AG Codes Associated to X(7)
We focus on the Klein quartic. We also use Elkies [E1, E2] as general references.
Let D_{∞} denote the reduced orbit of the point ∞, so \(\deg(D_{\infty})=G/N=24\), and let D=rD_{∞}. Let E=P_{1}+⋯+P_{ n } denote the sum of the remaining \({\mathbb{F}}(P)\)rational points of X, so D and E have disjoint support.
If C is as in (6.2.2), then map φ in (6.2.4) is injective. Since eval _{ E } is injective as well, the Gmodule structure of C is the same as that of L(D), which is known thanks to the Brauercharacter analog of the formula (6.7.1). See also Example 3 in [JK1].
With r=1, we expect that C=C(X,D,E) is a [56,22,32] code.^{8} With r=2, we expect that C=C(X,D,E) is a [56,46,8] code. With r=3, C=C(X,D,E) is a [56,56,1] code. In case r=1,2, eval _{ E } is injective, but when r=3, it is not. Indeed, \(\dim L(3D_{\infty})=70\). In each case, the dimension of C can be computed using SAGE (and Singular), but the minimum distance cannot.
Remark 26

Indeed, it is known more generally that, for an AG code constructed as above from a curve of genus g, \(n\leq\dim(C)+d(C)+g1\), where d(C) denotes the minimum distance (Theorem 3.1.1 in [TV] or Lemma 189 in Sect. 6.5 below). Therefore, as an AG code, the codes constructed above with r=1,2 are in some sense “best possible.”

In general, Sect. 4.1 of [TV] shows how to construct a family of “good” codes from the curves \(X=X_{N}'\) for prime N>5, with automorphism group G=PSL(2,p).
6.4.1 The Curves X_{0}(N) of Genus 1
Some examples, which we shall use later, are collected in Table 6.1.
When N=36, Sect. 4.3 in Rovira [Ro] gives y^{2}=x^{4}−4x^{3}−6x^{2}−4x+1, which is a hyperelliptic equation but not in Weierstrass form. When N=49, Sect. 4.3 in Rovira [Ro] gives y^{2}=x^{4}−2x^{3}−9x^{2}+10x−3, which is a hyperelliptic equation but not in Weierstrass form.
6.5 Some Estimates on AG Codes
This currently is an active field of research. An excellent general reference is the 2010 survey paper by Li [Li]. The survey by Li also presents recent work of Elkies, Xing, Li, Maharaj, Stichtenoth, Niederreiter, Özbudak, Yang, Qi and others, with more recent advances than described here. Below, some of the basic wellknown estimates are discussed.
Let g be the genus of a curve V=X, and let C=C(D,E,X) denote the AG code as constructed above in (6.2.2). If C has parameters [n,k,d], then the following lemma is a consequence of the Riemann–Roch theorem.
Lemma 189
Assume thatCis as above andDsatisfies\(2g2<\deg (D)<n\). Then\(k=\operatorname{dim} (C)=\deg (D)g+1\)and\(d\geq n\deg(D)\).

if g=0, then C is an MDS code,

if g=1, then n≤k+d≤n+1.
Proposition 190
WithCas in the previous lemma, we have\(\delta+R= {\frac{d}{n}} + {\frac{k}{n}} \geq1  \frac{g1}{n}\).
Theorem 186 is an explicit formula for the genus of the modular curve X_{0}(N) in terms of arithmetic data. Equation (6.3.6) is an estimate relating the genus of the modular curve with its number of points over a finite field. It may be instructive to plug these formulas into the estimate in Proposition 190 to see what we get. The formula for the genus g_{ N } of X_{0}(N) is relatively complicated but simplifies greatly when N is a prime number which is congruent to 1 modulo 12, say N=1+12m, in which case g_{ N }=m−1. For example, g_{13}=0. In particular, we have the following:
Corollary 191
The following result says that the Drinfeld–Vladut bound can be attained in the case q=p^{2}.
Theorem 192
(Tsfasman, Valdut, and Zink [TV], Theorem 4.1.52)
Letg_{ N }denote the genus ofX_{0}(N). IfNruns over a set of primes different from p, then the quotientsg_{ N }/X_{0}(N)(GF(p^{2})) associated to the modular curvesX_{0}(N) tend to the limit\(\frac{1}{p1}\).
More generally, if q=p^{2k}, then there is a family of Drinfeld curves X_{ i } over GF(q) yielding \(\alpha= \sqrt{q}1\) ([TV], Theorem 4.2.38, discovered independently by Ihara [I] at about the same time). In other words, the Drinfeld–Vladut bound is attained in the case q=p^{2k}.
6.6 Examples
Let X be an elliptic curve. This is a projective curve for which X(GF(q)) has the structure of an algebraic group. Let P_{0}∈X(GF(q)) denote the identity. Let P_{1},P_{2},…,P_{ n } denote all the other elements of X(GF(q)), and let A=aP_{0}, where 0<a<n is an integer.
Example 193
The following result is an immediate corollary of the results in [Sh1], see also Sect. 5.2.2 in [TV].
Theorem 194
(Shokrollahi)
 Ifa=2 andX(GF(q))≅C_{2}×C_{2} (whereC_{ n }denotes the cyclic group of ordern), then the codeC=C(D,E) is an [n,k,d]code (nis the length, kis the dimension, anddis the minimum distance) with$$d=nk+1\quad \mathit{and}\quad k=a.$$
 Assume that\(\gcd(n,a!)=1\). If\(a\not= 2\)orX(GF(q)) is not isomorphic to the Klein four groupC_{2}×C_{2}, thenC=C(D,E) is an [n,k,d]code (n is the length, kis the dimension, anddis the minimum distance) withand weight enumerator polynomial (see, for example, [MS] for the definition)$$k=a$$whereB_{ a }is given in [Sh1] and Sect. 3.2.2 in [TV].$$W_C(x)=x^n+\,\sum_{i=0}^{a1}\left(\begin{array}{c}n \\i\end{array}\right)(q^{ai}1)(x1)^i +B_a(x1)^a,$$
6.6.1 The Generator Matrix (According to Goppa)
This section uses the method of Goppa’s book [G1] to compute the generator matrices of some AG codes.
Example 195
Consider, for example, the case of p=7. Let D=mP_{∞} and E=P_{1}+⋯+P_{7}, and let C denote the onepoint AG code associated to X/GF(7) and these divisors D, E. These codes give rise to MDS codes in many cases.
The method used in Goppa’s Fermat cubic code example of [G1] (pp. 108–109) can be easily modified to yield analogous quantities for certain elliptic Goppa codes.
Example 196
An example of the generating matrix of a onepoint elliptic code associated to x^{3}+y^{3}=1 over GF(4) has been worked out in several places (for example, see Goppa’s book mentioned above or the books [SS], Sect. 3.3, [P], Sects. 5.3, 5.4, 5.7, and [Mo], Sect. 5.7.3).
6.7 Ramification Module of X(N)
The following result is due to Joyner and Ksir [JK1]. We use the notation of (6.1.1) and of the appendix Sect. 7.5 below.
Theorem 197
The formula for m_{ π }(N) is, though explicit, fairly complicated and will not be stated here (see [JK1] for details).
Open Problem 31
Suppose that X is a smooth projective curve with (a) genus greater than 1, (b) automorphism group G, and (c) defined over a field F with “bad” characteristic p (that is, p divides the order of G). Is there an analog of Theorem 197?
Is there an \({\mathbb{F}}[G]\)module decomposition of an arbitrary AG code analogous to (6.1.1)?
This motivates the following, as stated in the introduction.
Theorem 198
ForN>5 prime, the ramification module ofX(N) overX(1) has a natural ℚ[G]module structure if and only if\(N\equiv1 \pmod{4}\).
In this case, we can use formula (6.7.1) to compute the ramification module directly from the restricted representations (for details, see [JK1]). If \(N \equiv3 \pmod{4}\), the situation is more complicated, but we refer to [JK1] for details.
6.7.1 Example: N=7
The texts Fulton and Harris [FH] and Serre [Se2] are good general references for (complex) representations over finite groups.
The computer algebra system [GAP] computes information about PSL(2,N); one can use it to compute character tables, induced characters, and Schur inner products (the computations can also be done in SAGE). In the examples of X(7) and X(11) below, we use (6.1.1) to explicitly compute the Gmodule structure of the ramification module and some Riemann–Roch spaces in the case N=7.
Let \(\zeta=e^{\frac{2 \pi i}{7}}\), and let ℚ(q) denote the (quadratic) extension of ℚ by q=ζ+ζ^{2}+ζ^{4}. Let \(\mathcal{G}\) denote the Galois group of ℚ(q)/ℚ. Then \(\mathcal{G}\) acts on the irreducible representations G^{∗} by swapping the two threedimensional representations and fixing the others.
 If \(\theta_{1}\in H_{1}^{*}\), then \(\pi_{\theta_{1}}=\mathrm{Ind}_{H_{1}}^{G}\, \theta_{1}\) is 84dimensional. Moreover,$$\pi_{\theta_1}\cong \left\{\begin{array}{l@{\quad}l}2\pi_2\oplus2\pi_3\oplus2\pi_4\oplus4\pi_5\oplus4\pi_6,& \theta_1\not= 1,\\[2pt]\pi_1\oplus\pi_2\oplus\pi_3\oplus4\pi_4\oplus3\pi_5\oplus4\pi_6,& \theta_1=1.\end{array}\right.$$
 If \(\theta_{2}\in H_{2}^{*}\), then \(\pi_{\theta_{2}}=\mathrm{Ind}_{H_{2}}^{G}\, \theta_{2}\) is 56dimensional. Moreover,$$\pi_{\theta_2}\cong \left\{\begin{array}{l@{\quad}l}\pi_2\oplus\pi_3\oplus2\pi_4\oplus2\pi_5\oplus3\pi_6,& \theta_2\not= 1,\\[2pt]\pi_1\oplus\pi_2\oplus\pi_3\oplus2\pi_4\oplus3\pi_5\oplus2\pi_6,& \theta_2=1.\end{array}\right.$$
 If \(\theta_{3}\in H_{3}^{*}\) is a fixed nontrivial character, then \(\pi_{\theta_{3}}=\mathrm{Ind}_{H_{3}}^{G}\, \theta_{3}\) is 24dimensional. Moreover,$$\pi_{\theta_3^k}\cong \left\{\begin{array}{l@{\quad}l}\pi_3\oplus\pi_4\oplus\pi_5\oplus\pi_6, & k \text{ quad.\ nonres. }\ (\mathrm{mod}\ 7),\\[2pt]\pi_2\oplus\pi_4\oplus\pi_5\oplus\pi_6, & k \text{ quad.\ res. }\ (\mathrm{mod}\ 7),\\[2pt]\pi_1 \oplus\pi_5\oplus2\pi_6, & k\equiv0\ (\mathrm{mod}\ 7).\end{array}\right.$$
Footnotes
 1.
The expository paper [JS] discussed this in more detail from the computational perspective.
 2.
 3.
The space \(\mathbb{H}=\{ z \in\mathbb{C}\;\; \mathrm{Im}\,(z) > 0\}\) is also called the Poincaré upper half plane.
 4.
 5.
Type optional_packages() for the name of the latest version of this database. This loads both ClassicalModularPolynomialDatabase and AtkinModularPolynomialDatabase.
 6.In fact, if we write \(f(z)=\,\sum _{n=1}^{\infty}a_{n}q^{n}\), thenis the global Hasse–Weil zeta function of the elliptic curve C of conductor 11 with Weierstrass model y^{2}+y=x^{3}−x^{2} [Gel] (p. 252).$$\zeta_C(s)=\bigl(1p^{s}\bigr)^{1}\prod_{p\not= 11}\bigl(1a_pp^{s}+p^{12s}\bigr)^{1}$$
 7.
 8.
In other words, C has length 56, dimension 22 over \({\mathbb{F}}\), and minimum distance 32.
 9.
 10.
Recall Singleton’s bound: n≥d+k−1.
 11.
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