Modular Symbols over Function Fields of Elliptic Curves

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 $\mathrm{\infty }$ of $F$ and let $F_{\mathrm{\infty }}$ be the completion. The group $\bar{\mathrm{\Gamma }}={\mathrm{G}\mathrm{L}}_2(\mathcal{O})$ acts on $\mathcal{T}$, the Bruhat-Tits building of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(F_{\mathrm{\infty }})$. In this article, we construct the group of modular symbols over $\mathrm{\Gamma }$, a congruence subgroup of $\bar{\mathrm{\Gamma }}$. We prove that this space is given by an explicit set of generators and relations among them.