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Modular forms, de Rham cohomology and congruences. (English) Zbl 1337.14021
Authors’ 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.
Independently, J. Kibelbek [Proc. Am. Math. Soc. 142, No. 12, 4029–4038 (2014; Zbl 1309.11039)] has given an example of a space of weight 2 modular forms that does not admit a $$p$$-adic Hecke eigenbasis (also for nonordinary primes $$p$$).

##### MSC:
 14F40 de Rham cohomology and algebraic geometry 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations
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##### References:
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