Toroidal automorphic forms for function fields

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 χ is toroidal if and only if L(χ, 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. CONTENTS

Automorphic forms for elliptic function fields

“…By [9,Thm. 4.3], an Eisenstein series E( · , χ) is F ′ -toroidal if and only if L(χ, s)L(χχ F ′ , s) vanishes in s = 1/2.…”

“…Note that there is a digression in notation: all upper and lower indices K are suppressed in this paper. For instance, we write A for the space denoted by A K and H for the algebra denoted by H K in [8] and [9]. 5.6.…”

“…We denote the space of all F ′ -toroidal automorphic forms by A tor (F ′ ). This space is a H-invariant subspace of A, and it decomposes into a direct sum of the following three H-invariant subspaces: the space A 0,tor (F ′ ) of F ′ -toroidal cusp forms, the space E tor (F ′ ) that is generated by F ′ -toroidal derivatives of Eisenstein series and the space R tor (F ′ ) that is generated by toroidal derivatives of residual Eisenstein series (see sections 6 and 7 in [9] for the definition of the derivative of a (residual) Eisenstein series; we shall also give a brief description in paragraph 7.6).…”

“…By [9, Thm. 6.2], the two Eisenstein series in the theorem are F ′ -toroidal and by [9,Thm. 4.3 (ii)], E tor (F ′ ) is generated by these two linear independent functions.…”

“…In the rest of this paper, we will determine the space of toroidal automorphic forms for an elliptic curve X. This will be done based on theorems of [9] and calculations with L-series.…”