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.