Classical Modular forms and Galois representations

These notes are based on two lectures which were given by the author at the Workshop on Arithmetic Algebraic Geometry, I.I.T. Guwhati, India, fall 2008. They were intended to give a quick survey on classical complex modular forms and their l-adic Galois representations. Accordingly, the focus is rather on results and examples whereas proofs are mainly sketched or omitted. Nothing in these notes is new or original and most of the exposed material can be found in any standard textbook on the subject (such as [4], [6], [7]). Where this is not the case we have indicated precise references. 1 Classical modular forms 1.1 Definitions and notations Throughout these notes we fix two integers k,N ≥ 1. The symbol p will always denote a prime number. Let Γ(N) be the kernel of the reduction map SL2(Z)→ SL2(Z/NZ). By a congruence subgroup we mean a subgroup of SL2(Z) containing Γ(N) for suitable N . Prominent examples are given by the congruence subgroups of Hecke type SL2(Z) ⊇ Γ0(N) ⊇ Γ1(N) ⊇ Γ(N) where a matrix ( a b c d ) ∈ SL2(Z) lies in Γ0(N) (resp. Γ1(N)) iff c ≡ 0 (N) (resp. c ≡ 0 (N), a ≡ d ≡ 1 (N)). Let GL2 (R) be the subgroup of GL2(R) consisting of matrices with positive determinant. Let H be the complex upper half plane endowed with its natural GL2 (R)-action by linear fractional transformations: γ.z := az+b cz+d where a, b, c, d are the entries of γ as above and z ∈ H. If f is any C-valued function on H we put (f |k γ)(z) := det(γ)(cz + d)−kf(γ.z). This defines a GL2 (R)-action on the vector space of such functions, the socalled weight k action. Let Γ be a fixed congruence subgroup containing Γ(N). A modular form of weight k on Γ is a complex-valued function on H satisfying