## Columbia Mathematics Department Colloquium

On the parity of coefficients of modular forms

by

# Joël Bellaïche

## Brandeis University

Abstract:

Recently Nicolas and Serre have determined the structure of the Hecke

algebra acting on modular forms of level 1 modulo 2, and Serre has

conjectured the existence of a universal Galois representation over

this algebra. I'll explain the proof of this conjecture, and show how

that representation may be used to get new information on the parity

of the coefficients of modular forms of level 1

-- for example, on the parity of the values of the generalized Ramanujan's tau

functions. I'll also explain a still conjectural relation with the

partition function.

algebra acting on modular forms of level 1 modulo 2, and Serre has

conjectured the existence of a universal Galois representation over

this algebra. I'll explain the proof of this conjecture, and show how

that representation may be used to get new information on the parity

of the coefficients of modular forms of level 1

-- for example, on the parity of the values of the generalized Ramanujan's tau

functions. I'll also explain a still conjectural relation with the

partition function.