Perfectoid Drinfeld Modular Forms

Nicole, Marc-Hubert; Rosso, Giovanni

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.2, pp. 1045-1067.

Dans la première partie, nous revenons sur les courbes modulaires de Drinfeld associée à $\mathrm{GL}\left(2\right)$ en adoptant le point de vue perfectoïde, et nous montrons comment récupérer une portion (perfectisée) de la théorie des formes modulaires de Drinfeld $\pi$-adiques surconvergentes. Dans la seconde partie, nous présentons quelques problèmes ouverts portant sur les familles de formes modulaires de Drinfeld pour $\mathrm{GL}\left(n\right)$.

In the first part, we revisit Drinfeld modular curves associated to $\mathrm{GL}\left(2\right)$ from the perfectoid point of view, and we show how to recover (a perfectized) part of the theory of overconvergent $\pi$-adic Drinfeld modular forms. In the second part, we review open problems for families of Drinfeld modular forms for $\mathrm{GL}\left(n\right)$.

Classification : 11F33, 11F52, 11G09
Mots-clés : $p$-adic families, Drinfeld modular forms, perfectoid spaces

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