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Codes from Modular Curves

  • David JoynerEmail author
  • Jon-Lark Kim
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

One of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the so-called modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraic-geometric 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 Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

One of the most interesting class of curves, from the perspective of arithmetical algebraic geometry, are the so-called modular curves. Some of the most remarkable applications of algebraic geometry to coding theory arise from these modular curves. It turns out these algebraic-geometric 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 error-correcting 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)≅ℙ1. 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).

If D is nonspecial, then a formula in Borne [Bo] gives
$$ \bigl[L(D)\bigr]=(1-g_{X(1)})\bigl[k[G]\bigr]+\bigl[\mathrm{deg}_{eq}(D)\bigr]-\bigl[\tilde{\varGamma}_G\bigr].$$
(6.1.1)
Here gX(1) is the genus of X(1) (which is zero), square brackets denote the equivalence class of a representation of G, \(\deg _{eq}(D)\) is the equivariant degree of D, and \(\tilde{\varGamma}_{G}\) is the ramification module (these terms and notions are defined in Sect.  7.5 below, in [JK2], and in Borne’s paper [Bo]). This result is mostly stated here for later reference.

The G-module structure of L(D) is explicitly known if, in addition, \(N\equiv1\pmod{4}\). This is discussed further below.

As a corollary, it is an easy exercise now to compute explicitly the decomposition
$$H^1\bigl(X(N),k\bigr) = H^0\bigl(X(N),\varOmega^1\bigr)\oplus\overline{H^0\bigl(X(N),\varOmega^1\bigr)}=L(K)\oplus\overline{L(K)}$$
into irreducible G-modules, where K is a canonical divisor. This was discussed in [KP] (over k=ℂ) and [Sc] (over the finite field k=GF(N)). Indeed, Schoen observes that the multiplicities of the irreducible representations occurring in H1(X(N),k) can be interpreted in terms of the dimension of cusp forms and number of cusps on X(N).

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 error-correcting 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

If D is any divisor on X, then the Riemann–Roch space L(D) is a finite-dimensional F-vector space given by
$$ L(D)=L_X(D)= \bigl\{f\in F(X)^\times | \operatorname{div}(f)+D\geq0\bigr\}\cup\{0\},$$
(6.2.1)
where div (f) denotes the divisor of the function fF(X). These are the rational functions whose zeros and poles are “no worse than those specified by D.” Let (D) denote its dimension.
Let
$$D\in\operatorname{Div}(X)$$
be a divisor in X(F) stabilized by G whose support is contained in \(X({{\mathbb{F}}})\). Let \(P_{1},\ldots,P_{n}\in X({{\mathbb{F}}})\) be distinct points, and
$$E=P_1+\cdots+P_n\in\operatorname{Div}(X)$$
be stabilized by G. This implies that G acts on the set supp (E) by permutation. Assume that supp (D)∩supp (E)=∅. Choose an \({{\mathbb{F}}}\)-rational basis for L(D) and let \(L(D)_{{\mathbb{F}}}\) denote the corresponding vector space over \({{\mathbb{F}}}\). Let C denote the algebraic-geometric (AG) code
$$ C=C(D,E) = \bigl\{\bigl(f(P_1),\ldots,f(P_n)\bigr) \,|\, f\in L(D)_{{\mathbb{F}}}\bigr\}.$$
(6.2.2)
This is the image of \(L(D)_{{\mathbb{F}}}\) under the evaluation map
$$ \begin{array}{l}\operatorname{eval}_E:L(D)\rightarrow F^n,\\[4pt]f \longmapsto\bigl(f(P_1),\ldots,f(P_n)\bigr).\end{array}$$
(6.2.3)
The group G acts on C by gG sending
$$c=\bigl(f(P_1),\ldots,f(P_n)\bigr)\in C\longmapsto c'=\bigl(f\bigl(g^{-1}(P_1)\bigr),\ldots,f\bigl(g^{-1}(P_n)\bigr)\bigr),$$
where fL(D). First, we observe that this map, denoted φ(g), is well defined. In other words, if eval  E is not injective and c is also represented by f′∈L(D), so c=(f′(P1),…,f′(P n ))∈C, then we can easily verify \((f(g^{-1}(P_{1})),\ldots, f(g^{-1}(P_{n})))=(f'(g^{-1}(P_{1})),\ldots,f'(g^{-1}(P_{n})))\). (Indeed, G acts on the set supp (E) by permutation.) This map φ(g) induces a homomorphism of G into the permutation automorphism group of the code Aut (C), denoted
$$ \phi:G\rightarrow\operatorname{Aut}(C).$$
(6.2.4)

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

We say that the space L(D) separates points if for all points P,QX, f(P)=f(Q) (for all fL(D)) implies P=Q (see [Ha], Chap. II, Sect. 7). By Proposition IV.3.1 in Hartshorne [Ha], D very ample implies that L(D) separates points. In general, if L(D) separates points, then
$$\mathrm{Ker}(\phi)=\bigl\{g\in G \,|\, g(P_i)=P_i,\ 1\leq i\leq n\bigr\}.$$
It is known (proof of Proposition VII3.3, [Sti]) that if \(n=\deg(E)>2g+2\), then {gG | g(P i )=P i , 1≤in} is trivial. Therefore, if n>2g+2 and L(D) separates points, then φ is injective. Since (see Corollary IV.3.2 in Hartshorne [Ha]) \(\deg(D)>2g\) implies that D is very ample, the lemma follows. □

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:GP is an isomorphism (see, for example, [JKT]). In any case, using (6.2.4), we regard C as a G-module. In particular, the (bijective) evaluation map eval  E :L(D)→C in (6.2.3) is G-equivariant.

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 ℙ1? The analogy to keep in mind is that ℙ1 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(ℙ1), which correspond to localizations of a coordinate ring F[ℙ1] (see Moreno [Mo], Sect. 1.1).

We shall, for reasons of space, emphasize intuition over precision. What is a point? ℙ1 (as a set) may be thought of as the set of lines through the origin in affine space F2. We say that two points in F2−{(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 ℙ1.

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 ℙ1. In fact, Aut(ℙ1)=PGL(2,F).

Riemann–Roch Spaces

The only meromorphic functions on the Riemann sphere are the rational functions, so we focus on the F-valued rational functions on the ℙ1, denoted F(ℙ1). Let fF(ℙ1), so \(f(x)=\frac{p(x)}{q(x)}\) is a rational function, where x is a “local coordinate” on ℙ1, and p(x),q(x) are polynomials. In other notation,
$$F\bigl(\mathbb{P}^1\bigr)=F(x).$$
For example, a polynomial f(x) of degree n in x is an element of F(ℙ1) which has n zeros (by the fundamental theorem of algebra) and a pole of order n at “the point at infinity”, denoted ∞. (What this really means is that f(1/x) has a pole of order n at x=0.)

A divisor on1 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 ℙ1. 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)=P1+⋯+P n n∞ and supp (div (f))={P1,…,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 (ℙ1).

Let X=ℙ1, and let F(X) denote the function field of X (the field of rational functions on X).

Let ∞=[1:0]∈X denote the point at infinity. In this case, the Riemann–Roch theorem becomes
$$\ell(D)-\ell(-2\infty-D)=\deg(D)+1.$$
It is known (and easy to show) that if \(\deg(D)<0\), then (D)=0, and if \(\deg(D)\geq0\), then \(\ell(D)=\deg(D)+1\).

The Action of G on L(D)

Let F be algebraically closed, and let X=ℙ1/F, by which we mean ℙ1 with base field F, so the field of rational functions is F(ℙ1). In this case, Aut(X)=PGL(2,F).

The action ρ of Aut(X) on F(X) is defined by where f g (x)=(ρ(g)(f))(x)=f(g−1(x)).

Note that Y=X/G is also smooth and F(X) G =F(Y).

Of course, Aut(X) also acts on the group Div (X) of divisors of X, denoted g(∑ P d P P)=∑ P d P g(P), for g∈Aut(X), P a prime divisor, and d P ∈ℤ. It is easy to show that div (f g )=g(div (f)). Because of this, if div (f)+D≥0, then div (f g )+g(D)≥0 for all g∈Aut (X). In particular, if the action of G⊂Aut (X) on X leaves D∈Div (X) stable, then G also acts on L(D). We denote this action by
$$\rho:G \rightarrow\operatorname{Aut}\bigl(L(D)\bigr).$$
A basis for the Riemann–Roch space is explicitly known for ℙ1. For notational simplicity, let
$$m_P(x)=\left\{\begin{array}{l@{\quad}l}x, & P=[1:0]=\infty,\\[4pt](x-p)^{-1}, & P=[p:1].\end{array}\right.$$

Lemma 178

LetP0=∞=[1:0]∈Xdenote the point corresponding to the localizationF[x](1/x). For 1≤is, letP i =[p i :1] denote the point corresponding to the localization\(F[x]_{(x-p_{i})}\)forp i F. Let\(D=\sum_{i=0}^{s} a_{i} P_{i}\)be a divisor, a k ∈ℤ for 0≤ks.
  1. (a)
    IfDis effective, then
    $$\bigl\{1,m_{P_i}(x)^k \,| \, 1\leq k\leq a_i,0\leq i\leq s\bigr\}$$
    is a basis forL(D).
     
  2. (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 one-dimensional vector space) be any nonzero element. Then
    $$\bigl\{m_P(x)^{i}q(x)\, | \,0\leq i\leq d\bigr\}$$
    is a basis forL(D).
     
  3. (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 non-Abelian 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 representation-theoretic 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 half-plane3 ℍ, from a classical perspective. We shall recall them from both the classical and group-theoretical point of view. The latter perspective generalizes to higher-dimensional Shimura varieties [Del].

Arithmetic Subgroups

We assume that G=SL(2) is the group of 2×2 matrices with entries from an algebraically closed field Ω. In particular, the group of R-points of SL(2) for a subring RΩ with unit element 1 is defined by
$$\mathit{SL}(2, R) =\bigl\{ g \in M(2, R)\;|\; \det(g) =1 \bigr\},$$
where M(2,R) is the space of 2×2 matrices with entries from R. We now define congruence subgroups in SL(2,ℤ). Let SL(2,ℤ) be the subgroup of SL(2,ℝ) with integral matrices. Consider a natural number N, and let
$$\varGamma(N) = \left\{ \left[\begin{array}{c@{\quad}c}a & b\\c & d\end{array}\right] \in \mathit{SL}(2, \mathbb{Z})\,\Big|\,\begin{array}{l}a, d \equiv1\ (\mathrm{mod}\ N)\\[3pt]b , c \equiv0\ (\mathrm{mod}\ N)\end{array}\right\}.$$
We note that the subgroup Γ(N) is a discrete subgroup of SL(2,ℝ), which is called the principal congruence subgroup of levelN. Any subgroup of SL(2,ℤ) that contains the principal congruence subgroup is called a congruence subgroup.

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 ΓSL(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

Note that the group SL(2,ℝ) acts on ℍ by
$$g\cdot z = (az + b) (cz + d)^{-1} = \frac{az + b}{cz +d},$$
where z∈ℍ, Open image in new window .

We emphasize that the action of SL(2,ℝ) on ℍ is transitive, i.e., for any two points w1,w2∈ℍ, there is an element gSL(2,ℝ) such that w2=gw1. 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

Take Γ to be the Hecke subgroupΓ0(N) defined by
$$\varGamma_0(N) = \left\{ \left[\begin{array}{c@{\quad}c}a & b\\c & d\end{array}\right] \in \mathit{SL}(2, \mathbb{Z})\,\Big|\, c \equiv0 \ (\mathrm{mod}\, N)\right\}$$
for a natural number N. This is a congruence subgroup, and Y0(N)=Γ0(N)\ℍ is an arithmetic quotient. Such a quotient is not a compact subset, nor a bounded one; it is however a subset with finite measure (volume) under the non-Euclidean measure induced on the quotient from the group SL(2,ℝ) which is a locally compact group and induces the invariant volume element \(\frac{dx\wedge dy}{y^{2}}\), where x,y are the real and complex parts of an element z∈ℍ.

We now recall the basic ideas that turn an arithmetic quotient of the form Γ\ℍ into an algebraic curve. Let ΓSL(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 γΓ with the property that |tr (γ)|=2). Any two cusps x1,x2 such that δx2=x1 for an element δΓ 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 Y0(N)=Γ0(N)\ℍ by X0(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 half-plane 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 Γ0(N) for N≥3 are torsion free.

An Adelic View of Arithmetic Quotients

Consider the number field ℚ, the field of rational numbers. Let ℚ p be the (p-adic) completion of ℚ under the p-adic absolute value |⋅| p , where |a/b| p =pn whenever a,b are integers and \(a/b= p^{n}\,\prod_{\ell\not= p\ \mathrm{prime}}\ell^{e_{\ell}}\), n,e ∈ℤ. (Roughly speaking, ℚ p is a set of Laurent series in p whose coefficients belong to ℤ/pℤ.) Under the ordinary absolute value, the completion of ℚ is ℝ, also denoted ℚ=ℝ. These are topological fields (under the metric topology) and the ring of integers of ℚ p ,
$$\mathbb{Z}_p =\bigl \{x\in\mathbb{Q}_p \,|\, |x|\leq1\bigr\},$$
is a maximal compact open subring of ℚ p . The ring of adeles of ℚ is the commutative ring \({\mathbb{A}}\) that is given by the restricted direct product
$${\mathbb{A}}= \biggl\{ (x_\infty, x_2, \dots) \in\mathbb{R}\times\,\prod_p \mathbb{Q}_p\,\Big|\, \mbox{ all but a finite number of}\ x_p \in\mathbb{Z}_p\biggr\}.$$
In the product topology, \({\mathbb{A}}\) is a locally compact ring. If \({\mathbb{A}}_{f}\) denotes the set of adeles omitting the ℝ-component x, then \({\mathbb {A}}_{f}\) is called the ring of finite adeles, and we can write \({\mathbb{A}}= \mathbb{R}\times{\mathbb{A}}_{f}\). Under the diagonal embedding, ℚ is a discrete subgroup of \({\mathbb{A}}\).
We now consider the group G=GL(2). For a choice of an open compact subgroup \(K_{f} \subset G({\mathbb{A}}_{f})\), it is known that we can write the arithmetic quotient (which was originally attached to an arithmetic subgroup of ΓSL(2,ℚ)) as the following quotient:
$$Y(K_f) = G(\mathbb{Q}) \backslash\bigl[\mathbb{H}\times\bigl(G({\mathbb {A}}_f) /K_f\bigr)\bigr]=\varGamma\backslash H,$$
(6.3.1)
where
$$\varGamma= G(\mathbb{Q}) \cap G(\mathbb{R}) K_f. $$
(6.3.2)
Thus our arithmetic subgroup Γ is completely determined by K f . From now on we assume that K f has been chosen so that Γ is torsion free.

Definition 181

Let G=GL(2). To G is associated the Shimura variety Sh(G) as follows. Let N≥3 be a natural number. Let Γ(N) be the congruence subgroup of level N of SL(2,ℤ), and K=SO(2,ℝ) the orthogonal group of 2×2 real matrices A with determinant 1 satisfying t AA=I2, where I2 denotes the 2×2 identity matrix. Then
$$Y(N)=\varGamma(N) \backslash\mathbb{H}\cong\varGamma(N) \backslash G(\mathbb{R}) / K.$$
We call this the modular space of levelN. Let
$$K_f(N) = \biggl\{ g \in G\biggl( \prod_{p}\mathbb{Z}_p\biggr) \,\Big|\, g \equiv I_{2}\ (\mathrm{mod}\, N)\biggr\}$$
be the open compact subgroup of\(G({\mathbb{A}}_{f})\)of levelN. Then the modular space of level N can be written as
$$Y(N) \cong G(\mathbb{Q}) \backslash G({\mathbb{A}})/K K_f(N) =G(\mathbb{Q}) \backslash\bigl[\mathbb{H}\times\bigl(G({\mathbb{A}}_f)/K_f(N)\bigr)\bigr].$$
Thus,
$$X\bigl(K_f(N)\bigr) \cong Y(N).$$
Taking the projective limit over K f (N) by letting N get large (which means that K f (N) gets small), we see that \(\lim_{N} Y(N)= G(\mathbb{Q})\backslash[\mathbb{H}\times G({\mathbb{A}}_{f})]\). Then the (complex points of the) Shimura curveSh(G) associated to G=SL(2) is defined by
$$\mathit{Sh}(G)(\mathbb{C}) = G(\mathbb{Q}) \backslash\bigl[\mathbb{H}\times G({\mathbb{A}}_f)\bigr].$$
(6.3.3)
Many mathematicians have addressed the natural questions below.
  • What field are the curves X(N), X0(N) “naturally” defined over?

  • How can they be described explicitly using algebraic equations?

Regarding the first question, from the general theory of Shimura varieties we know that for each reductive group G defined over ℚ satisfying the axioms of Sect. 2.1.1 in [Del], there is an algebraic number field E=E G over which a Shimura variety Sh(G) is defined [Del]. In fact, the Shimura curves X(N) and X0(N) are regular schemes proper over ℤ[1/N] (more precisely, over Spec(ℤ[1/N])).4
Regarding the second question, it is possible to find a modular polynomialH N (x,y) of degree
$$\mu(N)=N \prod_{p|N}\biggl(1+{\frac{1}{p}}\biggr) $$
(6.3.4)
for which H N (x,y)=0 describes (an affine patch of) X0(N). Let
$$G_k(q)=2 \zeta(k)+2 {\frac{( 2 i\pi)^{k}}{(k-1)!}}\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^n,$$
where q=e2πiz,z∈ℍ, σ r (n)= ∑d|nd r , and let
$$\varDelta(q)=60^3G_4(q)^3-27\cdot140^2G_6(q)^2=q \prod_{n=1}^\infty\bigl(1-q^n\bigr)^{24}.$$
Define the j-invariant by (More details on Δ and j can be found, for example, in [Shim] or [Kob].) The key property satisfied by H N is H N (j(q),j(q N ))=0. It is interesting to note in passing that when N is such that the genus of X0(N) equals 0 (i.e., N∈{1,3,4,5,6,7,8,9,12,13,16,18,25} [Kn]), then this implies that (x,y)=(j(q),j(q N )) parameterizes X0(N). In general, comparing q-coefficients allows one to compute H N for relatively small values of N. (The SAGE command ClassicalModularPolynomialDatabase() loads a database which allows one to compute this expression.) However, even for N=11, some of the coefficients can involve one hundred digits or more. The cases N=2,3 are given, for example, in Elkies [E1].

Example 182

The following SAGE commands which illustrate this require David Kohel’s database database_kohel be loaded first.5
This is basically (20) in [E1].
The paper by Cohen [Co] determines the asymptotic size of the largest coefficient of H N (normalized to have leading coefficient equal to 1). She shows that the largest coefficient grows like Ncμ(N), where c>0 is a constant, and μ is as in (6.3.4). More practical equations for (some of) the X0(N) are given in Hibino and Murabayashi [HM], Shimura [ShimM], Rovira [Ro], Frey and Müller [FM], Birch [B], and Table 6.1 in Sect. 6.5 below.
Table 6.1

Models of genus 1 modular curves

Level

Discriminant

Weierstrass model

Reference

11

−11

y2+y=x3x2

[BK], Table 1, p. 82

14

−28

y2+xyy=x3

p. 391, Table 12.1 of [Kn]

15

15

y2+7 xy+2 y=x3+4 x2+x

p. 65, Table 3.2 of [Kn]

17

17

y2+3 xy=x3+x

p. 65, Table 3.2 of [Kn]

19

−19

y2+y=x3+x2+x

[BK], Table 1, p. 82

20

80

y2=x3+x2x

p. 391, Table 12.1 of [Kn]

21

−63

y2+xy=x3+x

p. 391, Table 12.1 of [Kn]

24

−48

y2=x3x2+x

p. 391, Table 12.1 of [Kn]

27

−27

y2+y=x3

p. 391, Table 12.1 of [Kn]

32

64

y2=x3x

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 X0(N)

In this section we recall some well-known though relatively deep results on X0(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.

First, some notation: let S2(Γ0(N)) denote the space of holomorphic cusp forms of weight 2 on Γ0(N)\H. Let T p :S2(Γ0(N))→S2(Γ0(N)) denote the Hecke operator defined by
$$T_pf(z)=f(pz)+\,\sum_{i= 0}^{p-1}f\biggl({\frac{z+i}{p}}\biggr),\quad z\in H.$$
Define \(T_{p^{k}}\) inductively by
$$T_{p^k}= T_{p^{k-1}}T_{p}-pT_{p^{k-2}},\quad T_1=1,$$
and define the modified Hecke operators \(U_{p^{k}}\) by
$$U_{p^k}=T_{p^{k}}-pT_{p^{k-2}},\quad U_p= T_p,$$
for k≥2. The Hecke operators may be extended to the positive integers by demanding that they be multiplicative.

Theorem 183

(“Congruence relation” of Eichler–Shimura [Mo], Sect. 5.6.7, or [St1])

Letq=p k , k>0 an integer. Ifpis a prime not dividingN, then
$$\mathrm{Tr}(T_p)=p+1-\big|X_0(N)\bigl(\mathit{GF}(p)\bigr)\big|.$$
More generally,
$$\mathrm{Tr}(T_q-pT_{q/p^2})=q+1-\big|X_0(N)\bigl(\mathit{GF}(q)\bigr)\big|.$$

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, S2(Γ0(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 S2(Γ0(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 a1=1) eigenform is the eigenvalue of T p (see, for example, Proposition 40 in Chap. III of [Kob]). If S2(Γ0(N)) is one-dimensional, then any element in that space f(z) is such an eigenform.

The modular curve X0(11) is of genus 1, so there is (up to a nonzero constant factor) only one holomorphic cusp form of weight 2 in S2(Γ0(11)) (see Theorem 186 below). There is a well-known 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.

Let q=e2πiz, z∈ℍ, and consider Dedekind’sη-function,
$$\eta(z)=e^{2\pi i z/24} \prod_{n=1}^\infty\bigl(1-q^n\bigr).$$
Then
$$f(z)=\eta(z)^2\eta(11z)^2q \prod_{n=1}^\infty\bigl(1-q^n\bigr)^{2}\bigl(1-q^{11n}\bigr)^{2}$$
is an element6 of S2(Γ0(11)). One can compute the q-expansion of this form using SAGE’s ModularForms(Gamma0(11),2) command:
$$f(z)=q-2\,{q}^{2}-{q}^{3}+2\,{q}^{4}+{q}^{5}+2\,{q}^{6}-2\,{q}^{7}\cdots .$$
For example, the above expansion tells us that Tr(T3)=Tr(U3)=−1. The curve X0(11) is of genus 1 and is isogenous to the elliptic curve C with Weierstrass model y2+y=x3x2. Over the field with p=3 elements, there are |X0(11)(GF(3))|=p+1−Tr(T p )=5 points in C(GF(3)), including ∞:
$$C\bigl(\mathit{GF}(3)\bigr)=\bigl\{[0, 0], [0, 2], [1, 0], [1, 2],\infty\bigr\}.$$
For this, one uses the SAGE commands
A plot of the real points of this elliptic curve is given in Fig. 6.1.
Fig. 6.1

The elliptic curve y2+y=x3x2 over ℝ

For a representation-theoretic discussion of this example, see [Gel], Sect. 14.

For an example of an explicit element of S2(Γ0(32)), see Koblitz [Kob] (Sect. 5 in Chap. II and (3.40) in Chap. III). For a remarkable theorem which illustrates how far this η-function 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.

Let k be an even positive integer, and let Γ be a congruence subgroup as in (6.3.2). Let S denote a complete set of representatives of G(ℚ)-conjugacy classes of ℝ-elliptic elements in Γ (ℝ-elliptic elements are those that are conjugate to an element of SO(2,ℝ), the orthogonal group). For γS, let w(γ) denote the cardinality of the centralizer of γ in Γ. If Open image in new window , then let θ γ ∈(0,2π) denote the element for which γ=r(θ γ ). Let τ m denote the image in \(G({\mathbb{A}}_{f})\) of the set of matrices in \(\mathit{GL}(2,{\mathbb{A}}_{f})\) having coefficients in \(\hat{\mathbb{Z}}= \,\prod_{p<\infty}\mathbb{Z}_{p}\) and determinant in \(m\hat{\mathbb{Z}}\). Consider the subspace S k (Γ)⊂L2(Γ\H) formed by the functions f satisfying
This is the space of holomorphic cusp forms of weight k on ℍ.
Let
$$\epsilon\bigl(\sqrt{m}\bigr)=\left\{\begin{array}{l@{\quad}l}1,& m\ \mathrm{is\ a\ square},\\0, & \mathrm{otherwise},\end{array}\right.$$
and let
$$\delta_{i,j}=\left\{\begin{array}{l@{\quad}l}1,& i= j,\\0, & \mathrm{otherwise}.\end{array}\right.$$

Theorem 185

(Eichler–Selberg trace formula)

Letk>0 be an even integer, andm>0 an integer. The trace ofT m acting onS k (Γ) is given by

Remark 24

Let k=2, m=p2, Γ=Γ0(N), and N→∞ in the above formula. It is possible to show that the Eichler–Selberg trace formula implies
$$\mathrm{Tr}(T_{p^2})= g\bigl(X_0(N)\bigr)+O(1) $$
(6.3.5)
as N→∞. The proof of this estimate (see [Mo], Chap. 5, or [LvdG], Sect. V.4) uses the explicit formula given below for g(X0(N))=dim (S2(Γ0(N)), which we shall also make use of later.

Theorem 186

(Hurwitz–Zeuthen formula [Shim])7

The genus ofX0(N) is given by
$$g\bigl(X_0(N)\bigr)=\operatorname{dim} \bigl(S_2\bigl(\varGamma_0(N)\bigr)\bigr)=1+{\frac{1}{12}}\mu(N)-{\frac{1}{4}}\mu_2(N)-{\frac{1}{3}}\mu_3(N)-\mu_\infty(N),$$
whereμis as in (6.3.4), and
$$\mu_\infty(N)= \sum_{d|N}\phi\bigl(\mathrm{gcd}(d,N/d)\bigr),$$
whereφis Euler’s totient function, and\(({\frac{\cdot}{p}})\)is Legendre’s symbol.
Estimate (6.3.5) and the Eichler–Shimura congruence relation imply as N→∞.

6.3.4 Modular Curves X(N)

Let H denote the complex upper half-plane, let ℍ=ℍ∪ℚ∪{∞}, and recall that SL(2,ℚ) acts on ℍ by fractional linear transformations. Let X(N) denote the modular curve defined over ℚ whose complex points are given by Γ(N)\ℍ, where
$$\varGamma(N)=\left\{ \left(\begin{array}{c@{\quad}c}a & b\\c & d\end{array}\right)\in \mathit{SL}_2(\mathbb{Z}) \,\big|\, a-1\equiv d-1\equiv b\equiv c\equiv0\ (\mathrm{mod}\ N)\right\}.$$
Throughout this paper, we will assume that N is prime and N>6. In this case, the genus of X(N) is given by the formula
$$g=1+\frac{(N-6)(N^2-1)}{24}.$$
For example, X(7) is of genus 3, and X(11) is of genus 26.
Let N be a prime, and, for j∈ℤ/Nℤ, let y j be variables satisfying for all a,b,c,d∈ℤ/Nℤ. These are Klein’s equations for X(N) (see Adler [A1] or Ritzenthaler [R1]).

Example 187

When N=7, this reduces to
$$y_1^3y_2-y_2^3y_3-y_3^3y_1=0,$$
the famous Klein quartic.
When N=11, the 20 equations which arise reduce to the 10 equations
$$\begin{array}{r}-y_1^2y_2y_3+y_2y_4y_5^2+y_3^2y_4y_5=0,\\[4pt]-y_1^3y_4+y_2y_4^3-y_3^3y_5=0,\\[4pt]-y_1y_3^3-y_1^3y_5+y_2^3y_4=0,\\[4pt]-y_1^2y_3y_4+y_1y_3y_5^2+y_2^2y_4y_5=0,\\[4pt]-y_1^2y_2y_5+y_1y_3y_4^2-y_2^2y_3y_5=0,\\[4pt]y_1^3y_2-y_3y_5^3-y_4^3y_5=0,\\[4pt]y_1y_5^3-y_2^3y_3+y_3^3y_4=0,\\[4pt]-y_1y_2^2y_4+y_1y_4^2y_5+y_2y_3^2y_5=0,\\[4pt]y_1y_2^3+y_2y_5^3-y_3y_4^3=0,\\[4pt]y_1y_2y_3^2+y_1y_4y_5^2-y_2y_3y_4^2=0.\end{array}$$

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).

Let
$$G=\mathit{PSL}_2(\mathbb{Z}/N\mathbb{Z})\cong\overline{\varGamma (1)}/\overline {\varGamma(N)},$$
where the overline denotes the image in PSL2(ℤ). This group acts on X(N). (In characteristic 0, see [Shim]; in characteristic >0, see [R1].) When N>2 is prime, |G|=N(N2−1)/2.

Definition 188

When X has good reduction to a finite field k and, in addition, the characteristic of k does not divide |G|, we say that is good.

If k is a field of good characteristic, the automorphism group of X(N) is known to be PSL(2,N) [BCG].

The action of G=SL2(ℤ/Nℤ) on the set of points of the projective curve defined by Klein’s equations is described in [R1] (see also [A2, R2]). The element Open image in new window sends (y j )j∈ℤ/NX(N) to (ρ(g)y j )j∈ℤ/NX(N), where
$$\rho(g)(y_j)=\sum_{t\in\mathbb{Z}/N\mathbb{Z}}\zeta^{b(aj^2+2jtc)+t^2cd}y_{aj+tc}$$
with ζ denoting a primitive Nth root of unity in k.

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 SL2(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 error-correcting codes.

Assume that 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 . (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.

First, we recall some notation and results from [TV]. Let A N =ℤ[ζ N ,1/N], where ζ N =e2πi/N, let K N denote the quadratic subfield of ℚ(ζ N ), and let B N =A N K N . There is a scheme X(N)/ℤ[1/N] which represents a moduli functor “parameterizing” elliptic curves E with a level N structure α N . There is a scheme X P (N)/ℤ[1/N] which represents a moduli functor “parameterizing” elliptic curves E with a “projective” level N structure β N . If P is a prime ideal in the ring of integers \(\mathcal{O}_{K_{N}}\) dividing , then the reduction of the form of X(N) defined over K N , denoted X(N)/P, is a smooth projective absolutely irreducible curve over the residue field k(P), with a PSL2(ℤ/Nℤ)-action commuting with the reduction. Similarly, with X(N) replaced by X P (N). Recall from Sect. 4.1.3 of [TV] that
$$k(P)=\left\{\begin{array}{l@{\quad}l}\mathit{GF}(\ell^2), & (\ell)=P,\\[2pt]\mathit{GF}(\ell), & (\ell)=PP'.\end{array}\right.$$
Let \(X'_{N}=X_{P}(N)/P\), and let
$$\psi'_N:X'_N\rightarrow X'_N/\mathit{PSL}_2(\mathbb{Z}/N\mathbb {Z})\cong\mathbb{P}^1$$
denote the quotient map. Let D denote the reduced orbit of the point ∞ (in the sense of Borne), so \(\deg(D_{\infty})=|G|/N\). Let D=rD, for r≥1. According to [TV], in general, this divisor is actually defined over GF(), not just k(P). Moreover, \(\deg(D)=r\cdot(N^{2}-1)/2\). Let E=P1+⋯+P n be the sum of all the supersingular points of \(X_{N}'\), and let
$$C=C\bigl(X'_N,D,E\bigr)=\bigl\{\bigl(f(P_1),\ldots,f(P_n)\bigr) \,| \, f\in L(D)\bigr\}$$
denote the AG code associated to \(X'_{N},D,E\). This is a G-module, via (6.2.4). Moreover, choosing r suitably yields a “good” family of codes with large automorphism group.

In fact, if D is “sufficiently large” (so, D is nonspecial, and both φ and eval  E are injective), then the Brauer-character analogs of formulas in Joyner and Ksir [JK1] give not only the G-module 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 \({\mathbb{F}}=\mathit{GF}(43)\). This field contains 7th roots of unity (ζ7=41), cube roots of unity (ζ3=36), and the square root of −7 (take \(\sqrt{-7}=6\)). Consider and Elkies [E2] points out that the matrix expression for ρ3 in terms of roots of unity can be found in Klein’s 1879 paper on (what is now known as) the Klein quartic.
It may be checked that these matrices preserve the form
$$\phi(x,y,z)=x^3y+y^3z+z^3x$$
over \({\mathbb{F}}\). They generate the subgroup GPSL2(7) of order 168 in \(\mathit{PGL}(3,{\mathbb{F}})\). The Klein curve x3y+y3z+z3x=0, denoted here by X, has no other automorphisms in characteristic 43, so \(G=\mathrm{Aut}_{\mathbb{F}}(X)\).

Let D denote the reduced orbit of the point ∞, so \(\deg(D_{\infty})=|G|/N=24\), and let D=rD. Let E=P1+⋯+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 G-module structure of C is the same as that of L(D), which is known thanks to the Brauer-character analog of the formula (6.7.1). See also Example 3 in [JK1].

The 80 points of \(X({\mathbb{F}})\) are The orbit of \((1:0:0)\in X({\mathbb{F}})\) under G is the following set of 24 points:
We evaluate each element of the Riemann–Roch space L(D) at a point in the complement of the above-mentioned orbit in the set of rational points \(X({\mathbb{F}})\).

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, \(n\leq\dim(C)+d(C)+g-1\), 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 X0(N) of Genus 1

It is known (see, for example, [Kn]) that a modular curve of level N, X0(N), is of genus 1 if and only if
$$N\in\{11,14,15,17,19,20,21,24,27,32,36,49\}.$$
In these cases, X0(N) is birational to an elliptic curve E having Weierstrass model of the form
$$y^{2}+ a_1xy+a_3y= {x}^{3}+a_2x^2 +a_4 x +a_6$$
with a1,a2,a3,a4,a6. If E is of the above form, then the discriminant is given by
$$\varDelta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6,$$
where The conductor9N of E and its discriminant Δ have the same prime factors. Furthermore, N|Δ [Kn, Gel].

Some examples, which we shall use later, are collected in Table 6.1.

When N=36, Sect. 4.3 in Rovira [Ro] gives y2=x4−4x3−6x2−4x+1, which is a hyperelliptic equation but not in Weierstrass form. When N=49, Sect. 4.3 in Rovira [Ro] gives y2=x4−2x3−9x2+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 well-known 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\(2g-2<\deg (D)<n\). Then\(k=\operatorname{dim} (C)=\deg (D)-g+1\)and\(d\geq n-\deg(D)\).

Consequently, k+dng+1. Because of Singleton’s inequality,10 we have:
  • if g=0, then C is an MDS code,

  • if g=1, then nk+dn+1.

The previous lemma also implies the following lower bound.

Proposition 190

([SS], Sect. 3.1, or [TV])

WithCas in the previous lemma, we have\(\delta+R= {\frac{d}{n}} + {\frac{k}{n}} \geq1 - \frac{g-1}{n}\).

Theorem 186 is an explicit formula for the genus of the modular curve X0(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 X0(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, g13=0. In particular, we have the following:

Corollary 191

LetX=X0(N), whereNis a prime number which is congruent to 1 modulo 12 and has the property thatXis smooth overGF(q). Then the parameters [n,k,d] of a Goppa code associated toXmust satisfy
$${\frac{d}{n}} + {\frac{k}{n}} \geq1 - \frac{{\frac{N-1}{12}}-2}{n}.$$
Based on Proposition 190, if one considers a family of curves X i with increasing genus g i such that
$$\lim_{i\rightarrow\infty}{\frac{|X_i(\mathit{GF}(q))|}{g_i}} = \alpha, $$
(6.5.1)
one can construct a family of codes C i with \(\delta(C_{i}) + R(C_{i}) \geq1 - \frac{1}{\alpha}\). It is known that \(\alpha\leq\sqrt{q}-1\) (this is the so-called Drinfeld–Vladut bound, [TV], Theorem 2.3.22).

The following result says that the Drinfeld–Vladut bound can be attained in the case q=p2.

Theorem 192

(Tsfasman, Valdut, and Zink [TV], Theorem 4.1.52)

Letg N denote the genus ofX0(N). IfNruns over a set of primes different from p, then the quotientsg N /|X0(N)(GF(p2))| associated to the modular curvesX0(N) tend to the limit\(\frac{1}{p-1}\).

More generally, if q=p2k, 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=p2k.

As a corollary to the above theorem, if p≥7, then there exists a sequence of AG codes C N over GF(p2) associated to a sequence of modular curves X0(N) for which (R(C N ),δ(C N )) eventually (for suitable large N) lies above the Gilbert–Varshamov bound in Theorem 21. This follows from comparing the Gilbert–Varshamov curve
$$\everymath{\displaystyle}\begin{array}{c}\bigl(\delta, f_q(\delta)\bigr),\\[4pt]f_q(\delta)=1- \delta\log_q\biggl({\frac{q-1}{q}}\biggr)-\delta\log_q(\delta)-(1-\delta)\log_q(1-\delta),\end{array}$$
with the curve \((\delta,\frac{1}{\sqrt{q}-1})\), q=p2.

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 P0X(GF(q)) denote the identity. Let P1,P2,…,P n denote all the other elements of X(GF(q)), and let A=aP0, where 0<a<n is an integer.

Example 193

Let X denote the elliptic curve of conductor 32 (and birational to X0(32)) with Weierstrass form y2=x3x. Let X(GF(p))={P0,P1,P2,…,P n }, where P0 is the identity, and let D=kP0 for some k>0, E=P1+⋯+P n . If p is a prime satisfying p≡3 (mod 4), then
$$\big|X\bigl(\mathit{GF}(p)\bigr)\big|=p+1$$
(Theorem 5, Sect. 18.4 in Ireland and Rosen [IR]). The parameters of the corresponding code C=C(D,E,X) satisfy n=p and d+kn, since g=1 by the above proposition. As we observed above, an AG code constructed from an elliptic curve satisfies either d+k−1=n (i.e., is MDS) or d+k=n. The result of Shokrollahi below implies that if, in addition, p>3 or k>2, then C is not MDS, and
$$n=p,\quad\quad d+k=p.$$

The following result is an immediate corollary of the results in [Sh1], see also Sect. 5.2.2 in [TV].

Theorem 194

(Shokrollahi)

LetX,P0,P1,…,P n ,D,E, be as above.
  • Ifa=2 andX(GF(q))≅C2×C2 (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
    $$d=n-k+1\quad \mathit{and}\quad k=a.$$
  • Assume that\(\gcd(n,a!)=1\). If\(a\not= 2\)orX(GF(q)) is not isomorphic to the Klein four groupC2×C2, thenC=C(D,E) is an [n,k,d]-code (n is the length, kis the dimension, anddis the minimum distance) with
    $$k=a$$
    and weight enumerator polynomial (see, for example, [MS] for the definition)
    $$W_C(x)=x^n+\,\sum_{i=0}^{a-1}\left(\begin{array}{c}n \\i\end{array}\right)(q^{a-i}-1)(x-1)^i +B_a(x-1)^a,$$
    whereB a is given in [Sh1] and Sect. 3.2.2 in [TV].

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 the hyperelliptic curve11X defined by y2=x p x over the field GF(p) with p elements. It is easy to see that
$$X\bigl(\mathit{GF}(p)\bigr)=\bigl\{P_\infty,(0,0),(1,0),\ldots,(p-1,0)\bigr\}$$
has exactly p+1 points, including the point at infinity, P. The automorphism group of this curve is a twofold cover of PSL(2,p) (see Göb [Go] for the algebraically closed case).

Consider, for example, the case of p=7. Let D=mP and E=P1+⋯+P7, and let C denote the one-point AG code associated to X/GF(7) and these divisors D, E. These codes give rise to MDS codes in many cases.

When m=2, we obtain a [7,2,6] code with weight enumerator 1+42x6+6x7. This code has automorphism group of order 252 and permutation group of order 42. When m=4, we obtain a [7,3,5] code with weight enumerator 1+126x5+84x6+132x7. This code has the same automorphism group and permutation group. It has the generator matrix in standard form
$$G=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 & 0 & 0 & 2 & 5 & 1 & 5\\0 & 1 & 0 & 1 & 5 & 5 & 2\\0 & 0 & 1 & 5 & 5 & 2 & 1\\\end{array}\right)$$
and check matrix
$$H=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}5 & 6 & 2 & 1 &0 &0 &0 \\2 & 2 & 2 & 0 &1 &0 &0 \\6 & 2 & 5 & 0 &0 &1 &0 \\2 & 5 & 6 & 0 &0 &0 &1\end{array}\right).$$

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

Let X denote the elliptic curve (of conductor N=19) which we write in homogeneous coordinates as
$$y^2z+yz^2=x^3+x^2z+xz^2.$$
Let φ(x,y,z)=x2+y2+z2, let Y denote the projective curve defined by φ(x,y,z)=0, and let D denote the divisor obtained by intersecting X and Y. By Bezout’s theorem, D is of degree 6. A basis for \(\mathcal{L}(D)\) is provided by the functions in the set
$$\mathcal{B}_D=\bigl\{ 1,x^2/\phi(x,y,z),y^2/\phi(x,y,z),z^2/\phi(x,y,z),xy/\phi(x,y,z),yz/\phi(x,y,z)\bigr\}.$$
(This is due to the fact that dim \(\mathcal{L}(D)=\deg(D)=6\) and the functions \(f\in\mathcal{B}_{D}\) “obviously” satisfy (f)≥−D.) We have which we write as P1, P2,…,P9. Consider the matrix
$$G=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0&0&0&1&1&1&1&1&1\\[-1pt]0&1&1&0&0&2&2&4&4\\[-1pt]1&0&1&4&2&2&1&4&1\\[-1pt]0&0&0&0&0&3&3&5&5\\[-1pt]0&0&6&0&0&5&4&3&2\\[-1pt]0&0&0&2&4&4&6&2&6\end{array}\right].$$
The first row of G gives the values of x2/φ(x,y,z) at {P i |1≤i≤9}. The other rows are obtained similarly from the other functions corresponding to the basis elements of \(\mathcal{L}(D)\): y2/φ(x,y,z), z2/φ(x,y,z), xy/φ(x,y,z), yz/φ(x,y,z). Performing Gauss reduction mod 7 puts this in canonical form:
$$G'=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1&0&0&0&0&0&0&4&4\\[-1pt]0&1&0&0&0&0&6&0&6\\[-1pt]0&0&1&0&0&0&1&3&4\\[-1pt]0&0&0&1&0&0&6&1&6\\[-1pt]0&0&0&0&1&0&1&3&5\\[-1pt]0&0&0&0&0&1&1&4&4\end{array}\right],$$
so this code also has minimum distance 3 and hence is only 1-error correcting. The corresponding check matrix is
$$H=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}0 &1 & 2& 1& 2& 2& 1& 0& 0\\[-1pt]3 &0 & 4& 6& 4& 3& 0& 1& 0\\[-1pt]3 &1 & 3& 1& 2& 3& 0& 0& 1\end{array}\right].$$

An example of the generating matrix of a one-point elliptic code associated to x3+y3=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

We have the following decomposition of the ramification module: wherem π (N) is an explicit multiplicity which satisfies\(\frac{N}{4}\leq m_{\pi}(N)\leq\frac{5N}{4}\)for all N.

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)?

If \(\tilde{\varGamma}_{G}\) has a ℚ[G]-module structure, it may be computed more simply. In this case, the formula for it is
$$ \tilde{\varGamma}_G=\bigoplus_{\pi\in G^*} \Biggl[\sum_{\ell=1}^L \bigl(\operatorname{dim} \pi-\operatorname{dim} \bigl(\pi^{H_\ell}\bigr)\bigr)\frac{R_\ell}{2} \Biggr]\pi,$$
(6.7.1)
where {H1,…,H L } represent the set of conjugacy classes of cyclic subgroups of G [JK2]. If \(\tilde{\varGamma}_{G}\) does not have a natural ℚ[G]-module structure, then the situation is more complicated, and we refer the reader to[JK1] for more details.

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 G-module structure of the ramification module and some Riemann–Roch spaces in the case N=7.

The equivalence classes of irreducible representations of PSL(2,7) are G={π1,π2,…,π6}, where

Let \(\zeta=e^{\frac{2 \pi i}{7}}\), and let ℚ(q) denote the (quadratic) extension of ℚ by q=ζ+ζ24. Let \(\mathcal{G}\) denote the Galois group of ℚ(q)/ℚ. Then \(\mathcal{G}\) acts on the irreducible representations G by swapping the two three-dimensional representations and fixing the others.

There are four conjugacy classes of nontrivial cyclic subgroups of G, whose representatives are denoted by H1 (order 2), H2 (order 3), H3 (order 7), H4 (order 4). We use GAP to compute the induced characters:
  • If \(\theta_{1}\in H_{1}^{*}\), then \(\pi_{\theta_{1}}=\mathrm{Ind}_{H_{1}}^{G}\, \theta_{1}\) is 84-dimensional. 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 56-dimensional. 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 24-dimensional. Moreover,
    $$\pi_{\theta_3^k}\cong \left\{\begin{array}{l@{\quad}l}\pi_3\oplus\pi_4\oplus\pi_5\oplus\pi_6, & k \text{ quad.\ nonres. }\ (\mathrm{mod}\ 7),\\[2pt]\pi_2\oplus\pi_4\oplus\pi_5\oplus\pi_6, & k \text{ quad.\ res. }\ (\mathrm{mod}\ 7),\\[2pt]\pi_1 \oplus\pi_5\oplus2\pi_6, & k\equiv0\ (\mathrm{mod}\ 7).\end{array}\right.$$
This data allows us to easily compute the ramification module using equations in [JK1]:
$$ \bigl[\tilde{\varGamma}_G\bigr] =[3\pi_2\oplus4\pi_3\oplus 6\pi_4\oplus7\pi_5\oplus8\pi_6].$$
(6.7.2)
Note that this is not Galois-invariant, because π2 and π3 have different multiplicities. A naïve computation of the ramification module, using (6.7.1), yields the following. For brevity, we represent m1[π1]+⋯+m6[π6] as (m1,…,m6). We compute, using GAP, the quantities Combining this with R1=R2=R3=1 and R4=0 in (6.7.2) gives This is impossible, and therefore we see that \(\tilde{\varGamma}_{G}\) does not have a ℚ[G]-module structure in this case.
Now we will use GAP to compute the equivariant degree and Riemann–Roch module for some example divisors. Any effective G-invariant divisor on X(7) will be nonspecial. Since X(1) is genus zero, for N=7, Borne’s formula (6.1.1) becomes If D1 is the reduced orbit of a point with stabilizer H1, then
$$\bigl[\deg_{eq}(D_1)\bigr]=[\pi_{\theta_1}] =[2\pi_2\oplus2\pi_3\oplus 2\pi_4\oplus4\pi_5\oplus4\pi_6]$$
and
$$\bigl[L(D_1)\bigr]=[\pi_1 \oplus2\pi_2\oplus\pi_3\oplus2\pi_4\oplus 4\pi_5\oplus4\pi_6].$$
If D2 is the reduced orbit of a point with stabilizer H2, then and If D3 is the reduced orbit of a point with stabilizer H3, then It follows that
$$\bigl[L(D_3)\bigr]=[\pi_1]-[\pi_3]+[\pi_3\oplus\pi_4\oplus\pi_5\oplus\pi_6]=[\pi_1\oplus\pi_4\oplus\pi_5\oplus\pi_6],$$
which is of dimension 22, and
$$\bigl[L(2D_3)\bigr]=[\pi_1]-[\pi_3]+2[\pi_3\oplus\pi_4\oplus\pi_5\oplus \pi_6]=[\pi_1\oplus\pi_3]+2[\pi_4\oplus\pi_5\oplus\pi_6],$$
which is of dimension 46.

Footnotes

  1. 1.

    The expository paper [JS] discussed this in more detail from the computational perspective.

  2. 2.

    For some introductions to this highly technical work of Langlands and Kottwitz, the reader is referred to Labesse [Lab], Clozel [Cl], and Casselman [Cas2].

  3. 3.

    The space \(\mathbb{H}=\{ z \in\mathbb{C}\;|\; \mathrm{Im}\,(z) > 0\}\) is also called the Poincaré upper half plane.

  4. 4.

    This result was essentially first proved by Igusa [Ig] (from the classical perspective). See also [TV], Theorem 4.1.48, and [Cas1] for an interesting discussion of what happens at the “bad primes,” and Deligne’s paper in the same volume as [Cas1].

  5. 5.

    Type optional_packages() for the name of the latest version of this database. This loads both ClassicalModularPolynomialDatabase and AtkinModularPolynomialDatabase.

  6. 6.
    In fact, if we write \(f(z)=\,\sum _{n=1}^{\infty}a_{n}q^{n}\), then
    $$\zeta_C(s)=\bigl(1-p^{-s}\bigr)^{-1}\prod_{p\not= 11}\bigl(1-a_pp^{-s}+p^{1-2s}\bigr)^{-1}$$
    is the global Hasse–Weil zeta function of the elliptic curve C of conductor 11 with Weierstrass model y2+y=x3x2 [Gel] (p. 252).
  7. 7.

    The genus formulas for X0(N) given in [Shim] and [Kn] both apparently contain a (typographical) error. The problem is in the μ2 term, which should contain a Legendre symbol \(({\frac{-4}{n}})\) instead of \(({\frac{-1}{n}})\). See, for example, [Ei] for a correct generalization.

  8. 8.

    In other words, C has length 56, dimension 22 over \({\mathbb{F}}\), and minimum distance 32.

  9. 9.

    The conductor is defined in Ogg [O1], but see also [Gel], Sect. I.2, or [Kn], p. 390.

  10. 10.

    Recall Singleton’s bound: nd+k−1.

  11. 11.

    When p=3 it is a model of a modular curve of level 32 (see Table 6.1). When p=7 this example arises in the reduction of X(7) in characteristic 7 [E2].

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentUS Naval AcademyAnnapolisUSA
  2. 2.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

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