Hall algebras and graphs of Hecke operators for elliptic curves

TL;DR

This paper explores the relationship between Hall algebras and graphs of Hecke operators over elliptic curves, providing algorithms to explicitly compute these graphs and their structure constants.

Contribution

It extends the understanding of Hecke operator graphs to elliptic curves and develops an explicit algorithm for their computation using Hall algebra structures.

Findings

Explicit algorithms for graphs of Hecke operators over elliptic curves

Calculation of structure constants in specific cases

Detailed analysis of rank two case

Abstract

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section.