Israel Journal of Mathematics

, Volume 194, Issue 2, pp 555–596 | Cite as

Toroidal automorphic forms for function fields

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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 x is toroidal if and only if L(x, 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 characteristic 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.

Keywords

Eisenstein Series Cusp Form Automorphic Form Riemann Hypothesis Automorphic Representation 
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Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Math. Dept.The City College of New YorkNew YorkUSA

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