Let $k =\mathbb{ F}_q$ be the finite field of $q$ elements and $E$ an elliptic curve over $k$. Let $F = k(E)$ be the function field over $E$ and let $\mathcal{ O} = k[E]$ be the ring of integers. We fix the place at $\infty$ of $F$ and let $F_{ \infty}$ be the completion. The group $\overline\Gamma = { \rm{ GL}}_2(\mathcal{ O})$ acts on $\mathcal{ T}$, the Bruhat-Tits building of ${ \rm{ PGL}}_2(F_{ \infty})$. In this article, we construct the group of modular symbols over $\Gamma$, a congruence subgroup of $\overline\Gamma$. We prove that this space is given by an explicit set of generators and relations among them.
Categories: math.NT