Automorphic forms for elliptic function fields

TL;DR

This paper provides explicit formulas for unramified Hecke operators on automorphic forms over elliptic function fields, using a geometric approach involving $ ext{P}^1$-bundles, and explores their implications for automorphic form spaces.

Contribution

It introduces a geometric method to compute Hecke operators on automorphic forms over elliptic function fields and analyzes their impact on the structure of these forms.

Findings

Formulas for unramified Hecke operators derived

Dimension of unramified cusp forms calculated

Toroidal automorphic forms characterized by zeta zeros

Abstract

Let $F$ be the function field of an elliptic curve $X$ over $\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of $\P^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the $\P^1$-bundles on $X$. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial $\P^1$-bundle. Further, we determine the space of unramified $F'$-toroidal automorphic forms where $F'$ is the quadratic constant field extension of $F$. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke $L$-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series $E(\blanc,s)$ where $s+1/2$ is a zero of the zeta function of $X$---with one possible exception in the case that $q$ is even and the class number $h$ equals $q+1$.