期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2020
卷号：117
期号：9
页码：4546-4558
DOI：10.1073/pnas.1912023117
出版社：The National Academy of Sciences of the United States of America
摘要：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 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 − n s ) ( 1 − q n − n s ) 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 where the sum is now over isomorphism classes of F q -rational semistable vector bundles V of degree 0 on X, is equal to ∏ k = 1 ∞ ζ X / F q ( s + k ) , and use this fact to prove the Riemann hypothesis for ζ X , n ( s ) for all n.
