Toroidal automorphic forms, Waldspurger periods and double Dirichlet series

TL;DR

This paper explores the structure of toroidal automorphic forms over arbitrary number fields, linking their properties to zeros of L-series and derivatives of Eisenstein series, using identities and non-vanishing results.

Contribution

It characterizes the space of toroidal automorphic forms in terms of Eisenstein series derivatives and cusp forms based on L-series zeros, extending previous work to general number fields.

Findings

Decomposition of toroidal automorphic forms into Eisenstein derivatives and cusp forms.

Connection between zeros of L-series and the structure of automorphic forms.

Application of double Dirichlet series methods for non-vanishing results.

Abstract

The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space spanned by all derivatives up to order n-1 of an Eisenstein series of weight s and class group character omega precisely if s is a zero of order n of the L-series corresponding to omega at s, and a space consisting of exactly those cusp forms the central value of whose L-series is zero. The proofs are based on an identity of Hecke for toroidal integrals of Eisenstein series and a result of Waldspurger about toroidal integrals of cusp forms combined with non-vanishing results for twists of L-series proven by the method of double Dirichlet series.