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 (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 is toroidal if and only if L(,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.