Modular forms, de Rham cohomology and congruences

Abstract:

In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for 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. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of “differentials of the second kind”. 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 Grössencharacter of the cyclotomic field $\mathbb {Q}(\zeta _N)$. Moreover, for $N=5$ the space does not admit a “$p$-adic Hecke eigenbasis” for (nonordinary) primes $p\equiv 2,3 \pmod {5}$, which provides a counterexample to Atkin and Swinnerton-Dyer’s original speculation.