Familles de formes modulaires de Drinfeld pour le groupe général linéaire

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.

In French. 35 pages. To appear in Transactions of the A.M.S