Automorphic forms for elliptic function fields

Let $F$ be the function field of an elliptic curve $X$ over ${\text{F}}_q$. 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 ${\mathrm{\P }}^1$-bundles on $X$. This allows a purely geometric approach, which involves, amongst others, a classification of the ${\mathrm{\P }}^1$-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 ${\mathrm{\P }}^1$-bundle. Further, we determine the space of unramified $F^{\mathrm{\prime }}$-toroidal automorphic forms where $F^{\mathrm{\prime }}$ 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 $E(\text{blanc},s)$ where $s+1\mathrm{/}2$ 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 $q+1$.