Une base explicite de symboles modulaires sur les corps de fonctions

Abstract

RésuméModular symbols for the subgroup Γ 0 (n) of GL 2 (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 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 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.