Geodesic
Cusps and $q$-expansion principles for modular curves at infinite level
Documenta mathematica, Tome 27 (2022), pp. 2385-2439 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/dm/x32
Classification : 11G18, 14G22, 14G35
Mots-clés : modular curve, cusps, infinite level, perfectoid 
@article{10_4171_dm_x32,
     author = {Ben Heuer},
     title = {Cusps and $q$-expansion principles for modular curves at infinite level},
     journal = {Documenta mathematica},
     pages = {2385--2439},
     year = {2022},
     volume = {27},
     doi = {10.4171/dm/x32},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/x32/}
}
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Ben Heuer. Cusps and $q$-expansion principles for modular curves at infinite level. Documenta mathematica, Tome 27 (2022), pp. 2385-2439. doi: 10.4171/dm/x32

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