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Transactions of the American Mathematical Society Series B

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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: 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 Serre’s 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 Dwork’s 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)