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Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  02 March 2021

Sean Howe
Sean Howe*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT, USA84112; E-mail: [email protected].

Abstract

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We construct a $(\mathfrak {gl}_2, B(\mathbb {Q}_p))$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $0$ of a sheaf on $\mathbb {P}^1$, landing in the compactly supported completed $\mathbb {C}_p$-cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $1$. For classical weight $k\geq 2$, the Verma has an algebraic quotient $H^1(\mathbb {P}^1, \mathcal {O}(-k))$, and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $\mathbb {P}^1$. Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology.

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e17
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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