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 Fq and any integer n ≥ 1 by ζ_X/_Fq,n(s) = ^{X |H0(X, V})r{0^}| q⁻deg(V)^s ((s) > 1), |Aut(V)| [V] where the sum is over isomorphism classes of Fq-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/Fq if n = 1, is a rational function of q^−s with denominator (1 − q^−ns)(1 − qⁿ−^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 1 X/_Fq(s) = ^X _[V_{] |}Aut(V)_| q⁻rank(V)^s ((s) > 0), where the sum is now over isomorphism classes of Fq-rational semistable vector bundles V of degree 0 on X, is equal to ^Q∞ _k=₁ ζ_X/_Fq(s + k), and use this fact to prove the Riemann hypothesis for ζ_{X ,n}(s) for all n.