  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.
