Cusps and q-expansion principles for modular curves at infinite level

2020/02/06 by Ben Heuer, Heuer, Ben
Mathematics · #11G18 #14G22 #14G35 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2002.02488

openalex publication_date 2020/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop an analytic theory of cusps for Scholze's p-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 Γ₁(p^∞) and the analogous locus of an infinite level perfected Igusa variety. We also prove various q-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 q-expansions.

Citations

• Perfectoid covers of abelian varieties

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