Perfectoid Drinfeld modular forms (Q2074004)

This paper under review consists of two parts: the first part revisits Drinfeld modular curves associated to \(\mathrm{GL}(2)\) from the perfectoid point of view and in the second part, the authors discuss several interesting open problems for families of Drinfeld modular forms for \(\mathrm{GL}(n)\). To fix the notation, let \({\mathcal C}\) be a projective smooth geometrically irreducible curve over \({\mathbb F}_{q}\) with function field \(F = {\mathbb F}_{q}({\mathcal C})\), \(\infty\) a fixed \({\mathbb F}_{q}\)-rational point and \(A\) the ring of regular functions outside \(\infty\). Fix an \(A\)-ideal \({\mathfrak N}\) which can be assumed to be principal and a prime ideal \({\mathfrak p}\) of norm \(q^{d}\), coprime to \({\mathfrak N}\). Let \(\pi\) be a generator of \({\mathfrak p}\) in \(A_{\mathfrak p}\). Following the perfectoid approach to Shimura varieties by \textit{P. Scholze} [Ann. Math. (2) 182, No. 3, 945--1066 (2015; Zbl 1345.14031)], in the first part of this note the authors show that there exists an infinite level Drinfeld modular curve which is a perfectoid space. To be precise, let \({\mathcal X}(\pi^{m})\) be the Drinfeld modular curve of full level \({\mathfrak p}^{m}\). the authors show (Theorem~2.18) that there exists a preperfectoid space \({\mathcal X}_{\infty} \sim \lim\limits_{\stackrel{\leftarrow}{m}} {\mathcal X}(\pi^{m})\), see [\textit{K. S. Kedlaya} and \textit{R. Liu}, Relative \(p\)-adic Hodge theory: foundations. Paris: Société Mathématique de France (SMF) (2015; Zbl 1370.14025)] for a definition of preperfectoid space. Then after extension of scalars to \(A_{\mathfrak p}[\![\pi^{1/p^{\infty}} ]\!][1/\pi]\), the inverse limit becomes perfectoid. Analogous to the case of Shimura varieties, \({\mathcal X}_{\infty}\) is equipped with a \(\mathrm{GL}_{2}(F_{\mathfrak p})\)-equivariant Hodge-Tate-Taguchi period map to \({\mathbb P}^{1}\). To show the existence of \({\mathcal X}_{\infty}\), as in Scholze's approach [loc. cit.], the authors first construct an anti-canonical tower of a strict neighborhood of the ordinary locus and then using the Hodge-Tate-Taguchi period map to extend it to the whole Drinfeld modular curve. Then, following [Doc. Math. 22, 191--262 (2017; Zbl 1455.11088)] \textit{P. Chojecki} et al. are able to to recover (a perfectized) part of the theory of overconvergent \(\pi\)-adic Drinfeld modular forms as functions on a subset of \({\mathcal X}_{\infty} \) satisfying the usual transformation property with respect to the action of \(\Gamma_{0}({\mathfrak p})\subset \mathrm{GL}_{2}(A_{\mathfrak p})\). More precisely, let \({\mathcal X}_{0}(\pi) (v)\) be a strict neighborhood of the ordinary locus of Drinfeld modules for which the Hasse invariant is bounded by \(v\). Then, given an analytic weight \(s \in {\mathbb Z}_{p}\) a line bundle \(\omega^{s}\) on the counterimage of \({\mathcal X}_{0}(\pi)(v)\) in \({\mathcal X}_{\infty}\) is constructed. The line bundle consists of functions satisfying the usual transformation formula and the authors show that this sheaf is the pullback from \({\mathcal X}_{0}(\pi)(v)\) of the perfection of the sheaf of overconvergent Drinfeld modular forms of weight \(s\) defined by the authors in another paper [Trans. Am. Math. Soc. 374, No. 6, 4227--4266 (2021; Zbl 1481.11050)]. In the second part, the authors discuss several problems related to topics including a conjectural \(r = t\) theorem; asking for a better definition of the Fredholm determinant in the non-noetherian context; asking about families of generalized modular forms for Anderson motives; the study of slopes à la Gouvêa-Mazur in higher rank; classicity in infinite slope (an example of a problem arising only for Drinfeld modular forms).