Modular forms, de Rham cohomology and congruences

**Speaker**: Dr. Matija Kazalicki, (University of Zagreb)

**Date**: Monday, November 24th

**Time**: 4:00pm

**Location**: Seminar Room, Ag 1.01

**Abstract**:

After reviewing Atkin and Swinnerton-Dyer's (ASD) speculation about existance of a "p-adic Hecke eigenbasis'' for the spaces of cusp forms for non-congruence subgroups, we briefly explain the connection of this phenomena with certain de Rham cohomology groups associated to modular forms. In our main result, we extend ASD congurences to weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups.

As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve X^N+Y^N=Z^N. We show that the Galois representation associated to this space is given by a Grossencharacter of the cyclotomic field \mathbb{Q}(\zeta_N). Moreover, for N=5 the space does not admit a "p-adic Hecke eigenbasis'' for (non-ordinary) primes p \equiv 2,3\pmod{5}, which provides a counterexample to Atkin and Swinnerton-Dyer's original speculation. This is joint work with Anthony J. Scholl (Cambridge).

**Series**: Algebra and Number Theory

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