Familles de formes modulaires de Drinfeld pour le groupe général linéaire

Abstract

Soient F F un corps de fonctions sur F q \mathbb {F}_q , A A l’anneau des fonctions régulières hors d’une place ∞ \infty et p \mathfrak {p} un idéal premier de A A . En premier lieu, nous développons la théorie de Hida pour les formes modulaires de Drinfeld de rang r r qui sont de pente nulle pour l’opérateur de Hecke U π \mathrm {U}_{\pi } convenablement défini. En second lieu, nous montrons en pente finie l’existence de familles de formes modulaires de Drinfeld variant continûment selon le poids. En troisième lieu, nous prouvons un résultat de classicité: une forme modulaire de Drinfeld surconvergente de pente suffisamment petite par rapport au poids est une forme modulaire de Drinfeld classique.

Perfectoid Drinfeld Modular Forms

“…Reminder on canonical subgroups and the Hodge-Tate-Taguchi decomposition. Fix a lift Ha of the Hasse invariant as in [28,Section 4] and let Ha be the truncated valuation of the Hasse invariant. For v ∈ Q ∩ [0, 1] let X (v) be a strict neighbourhood of the ordinary locus of Drinfeld modules for which Ha ≤ v. Fix a formal model X(v) for X (v) obtained as an open of in admissible blow-up of the formal model X of X .…”

“…Proof. Recall that X admits a proper model X over Spa(A p , A p ) equipped with the universal family of generalised Drinfeld modules [28,Théorème 2.6]. Given that ϕ corresponds to a K-point of X , by properness we can lift this to an O K -point of X and it will correspond to a generalised Drinfeld module ϕ over O K whose generic fiber is ϕ.…”

“…We apply [28, Lemme 5.2 and Proposition 5.4] to show that quotienting ϕ by a suitable subgroup of the p m -torsion will move the point into X(v); this is enough as the Tate modules of two isogenous Drinfeld modules are simply different lattices in T p (ϕ) ⊗ Ap F p . Note that if v is the truncated valuation of the Hasse invariant of ϕ, then one can use the explicit description of the canonical subgroup given in the proof of [28,Théorème 4.4] to see that its degree à la Fargues is 1 − v . Then Proposition 5.4 of loc.…”

“…In our previous paper [28], we explained how to adapt the eigenvariety machinery to the non-noetherian context of Drinfeld modular varieties associated to GL(n) for n ∈ N, including Hida theory in the form of an analogue of the Vertical Control Theorem, a continuous analogue of Coleman's finite slope families and a classicality theorem of overconvergent Drinfeld modular forms. This led to a variety of open questions, some intrinsic to the set-up of Drinfeld modular forms.…”

“…Then, given an analytic weight s ∈ Z p we would like, following [10], to define overconvergent Drinfeld modular forms in a concrete fashion as functions on a subset of X ∞ which satisfy the transformation formula (1.1) γ * f = (bz + d) −s f, for all γ ∈ Γ 0 (p) ⊂ GL 2 (A p ) and where z is a so-called fake Hasse invariant i.e., it is the pullback via the Hodge-Tate-Taguchi map of the coordinate z on P 1 . More precisely, let X 0 (π)(v) be a strict neighbourhood of the ordinary locus of Drinfeld modules for which the Hasse invariant is bounded by v. We construct a line bundle ω s on the counterimage of X 0 (π)(v) in X ∞ , consisting of functions satisfying the above formula (1.1) and we can only show that this sheaf is the pullback from X 0 (π)(v) of the perfection of the sheaf of overconvergent Drinfeld modular forms of weight s defined in [28].…”