Higher-rank zeta functions for elliptic curves.

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite field [Formula: see text] and any integer [Formula: see text] by[Formula: see text]where the sum is over isomorphism classes of [Formula: see text]-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function of [Formula: see text] if [Formula: see text], is a rational function of [Formula: see text] with denominator [Formula: see text] and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet series[Formula: see text]where the sum is now over isomorphism classes of [Formula: see text]-rational semistable vector bundles V of degree 0 on X, is equal to [Formula: see text] and use this fact to prove the Riemann hypothesis for [Formula: see text] for all n.