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
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