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 $\overline\Gamma = {\rm{GL}}_2(\mathcal{O})$ 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.