Let F be the function field of an elliptic curve X over \F_q. In this
paper, we calculate explicit formulas for unramified Hecke operators acting on
automorphic forms over F. We determine these formulas in the language of the
graph of an Hecke operator, for which we use its interpretation in terms of
¶1-bundles on X. This allows a purely geometric approach, which involves,
amongst others, a classification of the ¶1-bundles on X.
We apply the computed formulas to calculate the dimension of the space of
unramified cusp forms and the support of a cusp form. We show that a cuspidal
Hecke eigenform does not vanish in the trivial ¶1-bundle. Further, we
determine the space of unramified F′-toroidal automorphic forms where F′ is
the quadratic constant field extension of F. It does not contain non-trivial
cusp forms. An investigation of zeros of certain Hecke L-series leads to the
conclusion that the space of unramified toroidal automorphic forms is spanned
by the Eisenstein series E(\blanc,s) where s+1/2 is a zero of the zeta
function of X---with one possible exception in the case that q is even and
the class number h equals q+1.Comment: 26 page