A unipotent circle action on $p$-adic modular forms

Author:
Sean Howe

Journal:
Trans. Amer. Math. Soc. Ser. B **7** (2020), 186-226

MSC (2020):
Primary 11F33, 11F77

DOI:
https://doi.org/10.1090/btran/52

Published electronically:
November 5, 2020

MathSciNet review:
4170572

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Abstract | References | Similar Articles | Additional Information

Abstract: Following a suggestion of Peter Scholze, we construct an action of $\widehat {\mathbb {G}_m}$ on the Katz moduli problem, a profinite-Γ©tale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serreβs space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb {C}$. To construct the $\widehat {\mathbb {G}_m}$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we also prove a natural generalization of Dworkβs equation $\tau =\log q$ for extensions of $\mathbb {Q}_p/\mathbb {Z}_p$ by $\mu _{p^\infty }$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\widehat {\mathbb {G}_m}$ integrates the differential operator $\theta$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.

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Additional Information

**Sean Howe**

Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

MR Author ID:
936764

Email:
sean.howe@utah.edu

Keywords:
$p$-adic modular forms,
$p$-adic $L$-functions,
Igusa varieties,
$p$-divisible groups,
$p$-adic Hodge theory

Received by editor(s):
January 1, 2020

Received by editor(s) in revised form:
July 22, 2020

Published electronically:
November 5, 2020

Additional Notes:
The author was supported during the preparation of this work by the National Science Foundation under Award No. DMS-1704005.

Article copyright:
© Copyright 2020
by the author under
Creative Commons Attribution-Noncommercial 3.0 License
(CC BY NC 3.0)