preprint · arXiv (Cornell University) · 2020

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

Cited by 2Open access

Abstract

Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale 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 $\hat{\mathbb{G}}_m$-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 $τ=\log q$ for extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $μ_{p^\infty}$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\hat{\mathbb{G}}_m$ integrates the differential operator $θ$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.

Cite this paper

APA 7

Howe, S. (2020). A unipotent circle action on $p$-adic modular forms. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2003.11129

MLA 9

Howe, Sean. “A unipotent circle action on $p$-adic modular forms.” arXiv (Cornell University), 2020, https://doi.org/10.48550/arxiv.2003.11129.

Chicago

Howe, Sean. 2020. “A unipotent circle action on $p$-adic modular forms.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.2003.11129.

Harvard

Howe, S. (2020) ‘A unipotent circle action on $p$-adic modular forms’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.2003.11129.

BibTeX

@misc{howe2020unipotent,
  title = {{A unipotent circle action on $p$-adic modular forms}},
  author = {Howe, Sean},
  journal = {arXiv (Cornell University)},
  year = {2020},
  doi = {10.48550/arxiv.2003.11129}
}

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