Modular Symbols over Function Fields of Elliptic Curves

TL;DR

This paper constructs and explicitly describes the group of modular symbols over a congruence subgroup related to an elliptic curve's function field, extending the theory of modular symbols to this setting.

Contribution

It introduces the modular symbols over $ ext{GL}_2( ext{ring of integers})$ for elliptic curves over finite fields and provides explicit generators and relations.

Findings

Explicit generators for the modular symbols group.

Relations among generators are explicitly characterized.

Framework extends modular symbols to elliptic curve function fields.

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.