Toroidal automorphic forms for function fields

TL;DR

This paper investigates the structure of toroidal automorphic forms over function fields, linking derivatives of Eisenstein series to zeros of L-functions and characterizing the space's dimensions and representation properties.

Contribution

It establishes a precise criterion for when derivatives of Eisenstein series are toroidal based on L-function zeros and describes the dimension and representation-theoretic properties of the space of toroidal automorphic forms.

Findings

Derivatives of Eisenstein series are toroidal if and only if associated L-functions vanish to a certain order.

No non-trivial toroidal residues of Eisenstein series exist.

Dimension of the space of derivatives depends on genus, class number, and characteristic.

Abstract

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let $F$ be a global field. An automorphic form on $\GL(2)$ is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight $s$ is toroidal if $s$ is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The $(n-1)$-th derivative of a non-trivial Eisenstein series of weight $s$ and Hecke character $\chi$ is toroidal if and only if $L(\chi,s+1/2)$ vanishes in $s$ to order at least $n$ (for the "only if"-part we assume that the characteristic of $F$ is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals $h(g-1)+1$ if the characterisitc is not 2; in characteristic 2, the dimension is bounded from below by this number. Here $g$ is the genus and $h$ is the class number of $F$. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.