On graphs of Hecke operators

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 $n\ge 1.$ 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 $X={\mathbb{P}}^1({\mathbb{F}}_q).$

31 pages

Cited by in corpus (1)

• Automorphic forms for PGL(3) over elliptic function fields. Part 1: Graphs of Hecke operators