On ℓ-ADIC Representations and Congruences for Coefficients of Modular Forms

Abstract

The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let the unique cusp form of weight 12 for the full modular group be written

$$ \Delta = q\mathop \prod \limits_1^\infty (1 - q^n )^{24} = \sum\limits_1^\infty {\tau (n)q^n } $$

(1)

and note that the associated Dirichlet series has an Euler product

$$ \sum {\tau (n)n^{ - s} } = \prod (1 - \tau (p)p^{ - s} + p^{11 - 2s} )^{ - 1} $$

(2)

so that all the τ(n) are known as soon as the τ(p) are.

Many of the results described in these lectures were first obtained in correspondence between Serre and me during the last five years ; the disentanglement of our respective contributions is left to the reader, as an exercise in stylistic analysis. The dedication is from both of us.