Familles de formes modulaires de Drinfeld pour le groupe g\'en\'eral lin\'eaire

TL;DR

This paper develops a Hida theory for Drinfeld modular forms over function fields, constructs families of forms varying with weight, and proves a classicity result for small slope overconvergent forms.

Contribution

It introduces a Hida theory framework for rank r Drinfeld modular forms, constructs continuous families in the finite slope case, and establishes a classicity criterion for small slope forms.

Findings

Established Hida theory for Drinfeld modular forms of rank r.

Constructed families of forms varying with weight.

Proved classicity for small slope overconvergent forms.

Abstract

Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope zero for a suitably defined Hecke operator $\mathrm{U}_{\mathfrak{p}}$. Second, we show the existence in the finite slope case of families of Drinfeld modular forms varying continuously with respect to the weight. Finally, we show a classicity result: an overconvergent Drinfeld modular form of sufficiently small slope with respect to the weight is a classical Drinfeld modular form.