Introduction to Modular Forms

For the convenience of the reader we reproduce a few facts and definitions about modular forms, although these are covered in a number of other places. However, some normalizations of terminology are not completely standardized, so it seemed preferable to spend a few pages going over these facts.

Hecke operators are averaging operators similar to a trace. They operate on the space of modular forms. Let /be a modular form, f = ∑a _n q ⁿ , with associated Dirichlet series . It turns out that f is an eigenfunction for all Hecke operators if and only if the Dirichlet series has an Euler product. Such Euler products give relations among the coefficients, which show that they are multiplicative in n (i.e. a _mn =a _m a _n if m, n are relatively prime), and that they satisfy certain recurrence relation for prime power indices. The reader will find applications for these in Chapter VI, § 3. One of the basic problems of the theory is to organize into a coherent role the relations satisfied by these coefficients, and their effect on the arithmetic of number fields. The Hecke ones are in a sense the oldest. Later chapters touch on congruence relations. Manin [Man 4] found some which are much more hidden. The situation is very much in flux as this book is written.

We first define the Riemann surface obtained by taking the quotient of the upper half plane by a subgroup Г of SL ₂(Z), of finite index, and we show how to complete it to a compact Riemann surface X _Г . We then define modular forms and cusp forms for such subgroups. In a sense, these generalize the notion of differential form of the first kind on the Riemann surface defined above. Just as one can define a scalar product for differentials of the first kind on X _Г , one can extend the definition of this product to arbitrary cusp forms. The Hecke operators act essentially as a trace mapping, from one level to another. They act as Hermitian operators with respect to this scalar product.

The points at infinity (called cusps) on the quotient curves X _Г of the upper half plane turn out to be especially interesting.

In this chapter we describe the Eichler-Shimura theory already mentioned in the preceding chapter.

For the most part we have considered modular forms on SL ₂(Z). Incidentally in dealing with Hecke operators on such forms, we needed to pass to congruence subgroups. We want to return more systematically to modular forms on such subgroups. As already mentioned, there are three important such subgroups, which we called Γ₀(N), Γ₁(N) and Γ(N). We treat Γ₁(N) in some detail as an example. By conjugation, one can reduce the theory on Γ(N) to that of Γ₁(N).

Atkin-Lehner [A-L] showed how to construct in a natural way a basis for the space of modular forms of given level which are eigenfunctions for the Hecke operators prime to that level, satisfying the same formalism as for level 1. They worked on Γ₀(N). Miyake [Mi] extended this to the general case, including the modular forms in the sense of Langlands in the context of representation theory. See also Casselman [Ca]. More recently, Li [Li] reconsidered the matter in the style of Atkin-Lehner, following [A-L] very closely.

This chapter pertains both to Part II and Part III. It deals with periods of differentials of third kind. It reproduces with little change the arguments of Dedekind [Ded] for dlog η. These arguments are typical of those used in deriving the transformation law for more complicated modular forms, and are therefore included here for the convenience of the reader, to give him an early acquaintance with this formalism. We use the transformation constant in the normalization due to Rademacher [Rad] rather than the Dedekind symbol S(c, d) because it is more convenient.

The study of modular forms modulo p was originated by Swinnerton-Dyer [Sw D], who determined the structure of the algebra of modular forms mod p. Serre then showed how one can extend this theory in many ways, and in particular obtained results concerning the congruence properties mod higher powers of p for the coefficients of the q-expansions of modular forms. After laying the basic foundations for the q-expansions, we reproduce Swinnerton-Dyer’s results, and then some of Serre’s basic results, referring to his more extensive papers for the continuation of the theory.

Perhaps the most fascinating connection between modular forms and number theory is the way in which they are connected with the existence of non-abelian extensions. Shimura [Sh 3] first established a connection between coefficients of certain modular forms, and the traces of Frobenius elements in extensions K of Q whose Galois group has a representation in GL ₂ (F ₁ ), and K is the field of l-division points of the curve X₀(11), or in the Jacobian of X ₁ (N), Theorem 7.14 of [Sh 2].

We consider functions on a projective system which satisfy a compatibility relation. At each step, the sum of the values in a given fiber are equal to the values at the base point. Mazur isolated this notion [Maz 1], [Maz 2], [Ma-SwD], which turns out to be very prevalent in number theory. This followed the work of Iwasawa, working with group rings formed with a projective system of finite groups, so that the compatibility relation is merely a formulation independent of the group for the basic property of the natural homomorphism of group rings induced by a group homomorphism. Iwasawa’s work dealt with projective limits of ideal class groups, a topic pursued especially in papers of Coates with Sinnott, Lichtenbaum and Wiles, [CS 1], [CS 2], [C-Li], [CW]. For projective limits of divisor class groups in the modular function field, cf. the Kubert-Lang series [KL].

The oldest distribution is that defined by the Bernoulli polynomials, although of course their classical recurrence property was not called by that name.

This chapter is inserted for the convenience of the reader, showing how the values of the classical L-series and zeta function at the negative integers are the same as the values of the p-adic functions.

We have seen in Chapter X, § 3 that the modular form G _k has a q-expansion So the value of the ordinary zeta function appears as the constant term of a modular form (Eisenstein series, as it is called). This phenomenon, first exploited by Klingen [Kl 3] and Siegel [Si 4], has been highly developed by Serre [Se 5] and others. We give here more examples.