Toroidal Automorphic Forms for Function Fields

Abstract

The definition of a toroidal automorphic form is due to Don Zagier, who showed in a paper in 1979 that the vanishing of certain integrals of Eisenstein series over tori in GL(2) is related to the vanishing of the Riemann zeta function at the weight of the Eisenstein series; and thus a relation between the unitarizability of the space of unramified toroidal automorphic forms and the Riemann hypothesis. In the same paper, Zagier asked the question of what happens in the context of a function field of a curve over a finite field. In this thesis, we interpret automorphic forms over global function fields as functions on the vertices of certain graphs that can be described in terms of rank-2 bundles on this curve. By providing a reinterpretation of Hecke operators in terms of these graphs and the reinterpretation of cuspidality and toroidality, we are able to compute spaces of cusp forms and toroidal forms by looking at systems of linear equations. In particular, the question of the validity of the Riemann hypothesis can be reformulated as an eigenvalue problem.