Modular Symbols over Function Fields of Elliptic Curves

Abstract: 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 $\Gamma = {\rm{GL}}2(\mathcal{O})$ acts on $\mathcal{T}$, the Bruhat-Tits building of ${\rm{PGL}}_2(F{\infty})$. In this article, we use the action of $\Gamma$ on $\mathcal{T}$ to construct the space of modular symbols over $F$. We prove that this space is given by an explicit set of generators and relations among them.