Modular Forms

This theoretically-oriented section serves as a guide to the rest of the chapter. We recall the definition of modular forms, then briefly discuss q-expansions, Hecke operators, eigenforms, congruences, and modular symbols.

Fix positive integers N and k, let H denote the complex upper half plane. Denote by M_k(Gamma_1(N)) the space of modular forms on Gamma_1(N) of weight k. This is the complex vector space of holomorphic functions f : H -> C such that f((az + b/cz + d)) = (cz + d)^k f(z) qquad for all pmatrix(a&b cr c&d) in Gamma_1(N), and f(z) vanishes at each element of PP^1(Q)=Qunion{Infinity} (see, e.g., [DI95] for a more precise definition.)

A Dirichlet character is a homomorphism eps:(Z/NZ)^ * -> C^ * of abelian groups. Dirichlet characters are of interest because they decompose M_k(Gamma_1(N)) into more manageable chunks. If V is any complex vector space equipped with an action rho:(Z/NZ)^ * -> Aut(V) and eps is a Dirichlet character, we set V(eps) = { x in V : rho(a) x = eps(a)x all a in (Z/NZ)^ * }. The space M_k(Gamma_1(N)) is equipped with an action of (Z/NZ)^ * by the diamond-bracket operators < d >, which are defined as follows. Given /line(d) in (Z/NZ)^ *, choose a matrix pmatrix(a&b cr c &d) in Gamma_0(N) such that d mod N = /line(d). Then < d > f(z) = (cz + d)^(-k) f( (az + b/cz + d)). We call M_k(Gamma_1(N))(eps) the space of modular forms of weight k, level N, and character eps. This is the complex vector space of holomorphic functions f : H -> C such that f((az + b/cz + d)) = eps(a) (cz + d)^k f(z) qquad for all pmatrix(a&b cr c&d) in Gamma_1(N), and which vanish at the cusps. We let M_k([eps]) denote the direct sum of the spaces M_k(Gamma_1(N))(eps) as eps varies over the Gal(Qbar/Q)-conjugates of eps. It is unnecessary to specify the level because it is built into eps.

To summarize, for any integer k and positive integer N, there is a finite-dimensional C-vector space M_k(Gamma_1(N)). Moreover, M_k(Gamma_1(N)) = bigoplus_( all eps) M_k(Gamma_1(N))(eps) = bigoplus_( Gal(Qbar/Q) - class reps. eps) M_k([eps]). In Section Creation Functions, we describe how to create the spaces M_k(Gamma_1(N)) and M_k([eps]) in Magma, for any k >= 1, N >= 1, and character eps.

Let f be a modular form, and observe that since pmatrix(1&1cr 0&1) in Gamma_1(N), we have f(z)=f(z + 1). If we set q = ( exp)(2pi i z), there is a q-expansion representation for f: f = a_0 + a_1 q + a_2 q^2 + a_3 q^3 + a_4q^4 + ... . The a_n are called the Fourier coefficients of f. Magma{} contains an algorithm for computing a basis of q-expansions for any space of modular forms of weight k >= 2 (see Section Bases).

Fix a positive integer N and let M be a sum of spaces M_k(N, eps). Let q( - exp) : M -> C[[q]] denote the map that associates to a modular form f its q-expansion. One can prove that there is a basis f_1, ..., f_d of M that maps to a basis for the free Z-module q( - exp)(M) intersect Z[[q]]. See Section q-Expansions for how to compute such a basis in Magma. Let M_Z be the Z-module spanned by f_1, ..., f_d. For any ring R, we define the space of modular forms over R to be M_R = M_Z tensor_(Z) R. Thus M_R is a free R-module of rank d with basis the images of f_1, ..., f_d in M_Z tensor_(Z) R. The computation of M_R is discussed in Section Base Extension.

Any space M of modular forms is equipped with an action of a commutative ring T=Z[ ... T_n ... ] of Hecke operators. The computation of Hecke operators T_n and their characteristic polynomials is described in Section Operators.

An eigenform is a simultaneous eigenvector for every element of the Hecke algebra T. A newform is an eigenform that doesn't come from a space of lower level and is normalized so that the coefficient of q is 1. Section Newforms describes how to find newforms. Computation of the mod p reductions and p-adic and complex embeddings of a newform is described in Section Reductions and Embeddings.

Computation of congruences is discussed in Section Congruences.

Modular symbols are closely related to modular forms. See Section Modular Symbols for the connection between the two.