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

TL;DR

This paper develops explicit computational methods for automorphic forms on PGL(3) over elliptic function fields, focusing on describing the graphs of Hecke operators and establishing dualities for all n and degrees.

Contribution

It provides explicit formulas for Hecke operator graphs for GL(3), describes key components of these graphs, and proves dualities applicable to all n, degrees, and function fields.

Findings

Complete description of degree 1 Hecke graphs for GL(3)

Description of even component of degree 2 Hecke graphs for GL(2)

Dualities for Hecke operators for all n and degrees

Abstract

This is a first part of a series of papers in which we develop explicit computational methods for automorphic forms for GL(3) and PGL(3) over elliptic function fields. In this first part, we determine explicit formulas for the action of the Hecke operators on automorphic forms on GL(2) and GL(3) in terms of their graphs. Our primary result consists in a complete description of the graphs of degree 1 Hecke operators for GL(3). As complementary results, we describe the 'even component' of the graphs of degree 2 Hecke operators for GL(2) and the 'neighborhood of the identity' of the graphs of degree 2 Hecke operators for GL(3). In addition, we establish two dualities for Hecke operators for GL(n) and PGL(n), which hold for all n, all degrees and all function fields.