Toroidal automorphic forms for some function fields

TL;DR

This paper characterizes toroidal automorphic forms over certain function fields, demonstrating their connection to zeta function zeros and providing an automorphic proof of the Riemann hypothesis for these cases.

Contribution

It explicitly computes the space of toroidal automorphic forms for specific function fields and links their structure to the zeros of zeta functions, offering a new proof of the Riemann hypothesis in this context.

Findings

The space of toroidal automorphic forms has dimension g.

It is spanned by Eisenstein series corresponding to zeros of zeta functions.

Provides an automorphic proof of the Riemann hypothesis for these function fields.

Abstract

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL_2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL_2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g zero or one, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an "automorphic" proof for the Riemann hypothesis for the zeta function of those curves.