Toroidal automorphic forms for some function fields

Details Toroidal automorphic forms for some function fields Toroidal automorphic forms for some function fields

2007-10-16

arxiv.org/abs/0710.2994v2

Mathematics

Number Theory

7 pages, 2 figures; v2: minor corrections

Scientific paper

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL_2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL_2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g zero or one, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an "automorphic" proof for the Riemann hypothesis for the zeta function of those curves.

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