Modular forms, de Rham cohomology and congruences
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- by Matija Kazalicki and Anthony J. Scholl PDF
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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.References
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Additional Information
- Matija Kazalicki
- Affiliation: Department of Mathematics, University of Zagreb, Bijenicka cesta 30, Zagreb, Croatia
- MR Author ID: 837906
- 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
- MR Author ID: 156735
- Email: a.j.scholl@dpmms.cam.ac.uk
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7097-7117
- MSC (2010): Primary 14F40, 11F33, 11F80
- DOI: https://doi.org/10.1090/tran/6595
- MathSciNet review: 3471086