Abstract
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 non-split 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 ⩽ 1 , 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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