Geodesic
Cusps and $q$-expansion principles for modular curves at infinite level
Documenta mathematica, Tome 27 (2022), pp. 2385-2439.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\Gamma_1(p^\infty)$ 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.
Classification : 11G18, 14G35, 14G22
Keywords: modular curve, infinite level, cusps, perfectoid 
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     author = {Heuer, Ben},
     title = {Cusps and \(q\)-expansion principles for modular curves at infinite level},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a7/}
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Heuer, Ben. Cusps and \(q\)-expansion principles for modular curves at infinite level. Documenta mathematica, Tome 27 (2022), pp. 2385-2439. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a7/