Une base explicite de symboles modulaires sur les corps de fonctions

Abstract

Modular symbols for the congruence subgroup ${\Gamma }_0(\mathfrak{n})$ of $G{L}_2({\mathbf{F}}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for ${\mathbf{F}}_q(T)$. The main statement also provides a key-step for a result towards the uniform boundedness conjecture for Drinfeld modules of rank $2$.

arXiv commentJournal für die reine und angewandte Mathematik (2014) à paraître, in French