Modular symbols for $\mathbf{F}_q(T)$ and Manin's presentation

Modular symbols are a useful theoretical and computational tool for
modular forms. In 1992, Teitelbaum introduced modular symbols for the
function field $\mathbf{F}_q(T)$. They have a presentation with a finite number of
generators and their relations, which is formally similar to Manin's
presentation of ``classical'' modular symbols for the rational field $\mathbf{Q}$.

In this talk, I will explain how the finite presentation of Teitelbaum's
modular symbols can be solved explicitly in a rather general case for
the congruence subgroup $\Gamma_0({\eufm n})$ of $GL_2(\mathbf{F}_q[T])$. This statement has
no known counterpart for classical modular symbols, where the
presentation has to be solved on a case by case basis. As an
application, we will see a non-vanishing statement for $L$-functions of
certain automorphic cusp forms for $\mathbf{F}_q(T)$.