Modular Symbols over Function Fields of Elliptic Curves

TL;DR

The paper develops a finite presentation of modular symbols for function fields of elliptic curves by embedding the problem into the Bruhat-Tits tree $\mathcal{T}$ and exploiting the $\Gamma={\rm GL}_2(\mathcal{O})$-action to control the combinatorics of cusps and ends via a finite subtree.It introduces four reduced symbol types—$e$-, $s$-, $o$-, and $ns$-symbols—and proves that any modular symbol is a finite sum of these reduced symbols, with all relations among symbols reducible to a finite, type-by-type set of relations.The main technical contribution is a hierarchy of finite relations (two-term, three-term, and Y-term) for each symbol type, together with a balancing argument on edges of the tree that enables an inductive elimination of symbols and a complete description of $H^{BM}_1(\Gamma\backslash\mathcal{T},\mathbb{Z})$ in terms of reduced-symbol data.This yields a tractable framework for computing modular symbols and Hecke actions on congruence subgroups, with potential applications to automorphic forms over the function field $GL_2(F)$ and explicit arithmetic of the associated modular 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 $Γ= {\rm{GL}}_2(\mathcal{O})$ acts on $\mathcal{T}$, the Bruhat-Tits building of ${\rm{PGL}}_2(F_{\infty})$. In this article, we use the action of $Γ$ 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.

Figures (14)

• Figure 1: The Tree $\mathcal{T}$ for $q=3$.
• Figure 2: The Subtree $\mathcal{S}$ for $y^2=x^3+x-1$ over $\mathbb{F}_3$.
• Figure 3: An example of the labeling for finitely many vertices of $\mathcal{T}$ for $y^2=x^3+x-1$ over $\mathbb{F}_3$.
• Figure 4: The sequences of the four reduced symbols for $y^2=x^3+x-1$ over $\mathbb{F}_3$.
• Figure 5: An example of the four reduced symbols on $\mathcal{T}$ for $y^2=x^3+x-1$ over $\mathbb{F}_3$.

Theorems & Definitions (33)

• Theorem 1.1: Main Result 1, Theorem \ref{['uni sum']}
• Theorem 1.2: Main Result 2, Theorem \ref{['main result']}
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8