On graphs of Hecke operators

TL;DR

This paper explores the properties of graphs associated with Hecke operators acting on automorphic forms over function fields, providing descriptions, conditions for connections, and explicit calculations for the projective line.

Contribution

It introduces the first properties of Hecke operator graphs for all ranks, linking them to coherent sheaves and offering a method to compute these graphs explicitly.

Findings

Description of Hecke graphs via coherent sheaves

Numerical condition for vertex connectivity

Explicit calculations for the projective line

Abstract

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms. Let $X$ be a curve over $\mathbb{F}_q$, $F$ its function field and $\mathbb{A}$ the adele ring of $F$. In this paper we will exhibit the first properties for the graph of Hecke operators for $\mathrm{GL}_n(\mathbb{A}),$ for every $n \geq 1.$ This includes a description of the graph in terms of coherent sheaves on $X.$ We provide a numerical condition for two vertices to be connected by an edge. Moreover, we describe how to calculate these graphs in the case of the projective line $X = \mathbb{P}^1(\mathbb{F}_q).$