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 F-q and any integer n >= 1 by zeta X/Fq,n(s) = Sigma([V])vertical bar H-0(X, V)\{0}vertical bar/vertical bar Aut(V) q-deg(V)s (R(s) > 1), where the sum is over isomorphism classes of F-q-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 X/F-q if n = 1, is a rational function of q(-s) with denominator (1 - q(-ns))(1 - q(n-ns)) 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 3(X)/F-q (s) =Sigma(vertical bar V vertical bar) 1/vertical bar Aut(V) q(-rank(V)s) (R(s) > 0), where the sum is now over isomorphism classes of F-q-rational semistable vector bundles V of degree 0 on X, is equal to Pi(infinity)(k=1) zeta X/F-q (s + k), and use this fact to prove the Riemann hypothesis for zeta(X,n)(s) for all n.