Modular forms, de Rham cohomology and congruences

2013/01/24 by Matija Kazalicki, Kazalicki, Matija, A. J. Scholl +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1301.5876

openalex publication_date 2013/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Grossencharacter of a cyclotomic field \Q(ζ_N). Moreover, for N=5 the space does not admit a "p-adic Hecke eigenbasis" for (non-ordinary) primes p≡ 2,3 \pmod5, which provides a counterexample for original Atkin and Swinnerton-Dyer speculaction (see [2], [7], [8]).

Cited by

• Atkin and Swinnerton-Dyer congruences for meromorphic modular forms
• Elliptic curves and Fourier coefficients of meromorphic modular forms

Related

• On Atkin and Swinnerton-Dyer congruence relations (3)
• Atkin and Swinnerton-Dyer congruences for meromorphic modular forms
• Congruences between modular forms and related modules
• Non-ordinary primes and congruences for weakly holomorphic modular forms of level 2
• Modular forms of CM type mod ℓ