### Abstract

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 − 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 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.

Original language | English |
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Pages (from-to) | 4546-4558 |

Number of pages | 13 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 117 |

Issue number | 9 |

DOIs | |

Publication status | Published - Mar 3 2020 |

### All Science Journal Classification (ASJC) codes

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*117*(9), 4546-4558. https://doi.org/10.1073/pnas.1912023117