Toroidal automorphic forms for some function fields (Q1019842)

If \(F\) denotes the function field of some smooth projective curve \(X\) over a finite field, and \(F\) is not rational, has class number 1, has a rational place and genus greater than or equal to 1, then by the Hasse-Weil theorem \(X\) is necessarily one of three elliptic curves which can be given explicitly. For such function fields, the authors compute the spaces \(T_F\) of automorphic forms on \(\mathrm{GL}_2\) with the property that all right translates integrate to zero over all non-split tori in \(\mathrm{GL}_2\) -- i.e., the ``toroidal automorphic forms'' of the title. In each case, the spaces \(T_F\) are one-dimensional, spanned by Eisenstein series of weight equal to a zero of the zeta function of \(F\). In proving this, the authors establish various facts which they use to give a new proof of the Riemann hypothesis for these function fields.