Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL2 is toroidal if all its right translates integrate to zero over all non-split tori in GL2, 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.