Cusps and -expansion principles for modular curves at infinite level

  • Ben Heuer

    Institut für Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany
Cusps and $q$-expansion principles for modular curves at infinite level cover
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Abstract

We develop an analytic theory of cusps for Scholze's -adic modular curves at infinite level in terms of perfectoid parameter spaces for Tate curves. As an application, we describe a canonical tilting isomorphism between an anticanonical overconvergent neighbourhood of the ordinary locus of the modular curve at level and the analogous locus of an infinite level perfected Igusa variety. We also prove various -expansion principles for functions on modular curves at infinite level, namely that the properties of extending to the cusps, vanishing, coming from finite level, and being bounded, can all be detected on -expansions.

Cite this article

Ben Heuer, Cusps and -expansion principles for modular curves at infinite level. Doc. Math. 27 (2022), pp. 2385–2439

DOI 10.4171/DM/X32