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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Modular forms, de Rham cohomology and congruences


Authors: Matija Kazalicki and Anthony J. Scholl
Journal: Trans. Amer. Math. Soc. 368 (2016), 7097-7117
MSC (2010): Primary 14F40, 11F33, 11F80
DOI: https://doi.org/10.1090/tran/6595
Published electronically: December 22, 2015
MathSciNet review: 3471086
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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.


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Additional Information

Matija Kazalicki
Affiliation: Department of Mathematics, University of Zagreb, Bijenicka cesta 30, Zagreb, Croatia
Email: mkazal@math.hr

Anthony J. Scholl
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: a.j.scholl@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/tran/6595
Received by editor(s): April 22, 2013
Received by editor(s) in revised form: April 28, 2014, July 21, 2014, and September 4, 2014
Published electronically: December 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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