Graduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan;
Max Planck Institute for Mathematics, 53111 Bonn, Germany;
Proc Natl Acad Sci U S A. 2020 Mar 3;117(9):4546-4558. doi: 10.1073/pnas.1912023117. Epub 2020 Feb 18.
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.
在 L.W. 的早期工作中，对于有限域上的任意光滑曲线 X 和任意整数 [Formula: see text]，通过[Formula: see text]定义了一个非阿贝尔 ζ 函数，其中和是 X 上秩为 n 的 [Formula: see text]-有理半稳定向量丛 V 的同构类，其度可被 n 整除。如果 [Formula: see text]，这个函数与通常的 [Formula: see text]的阿廷 ζ 函数一致，是 [Formula: see text]的有理函数，分母为 [Formula: see text]，并且据推测满足黎曼假设。在本文中，我们详细研究了亏格为 1 的曲线的情况。我们证明，在这种情况下，狄利克雷级数[Formula: see text]其中和现在是 X 上度为 0 的 [Formula: see text]-有理半稳定向量丛 V 的同构类，等于 [Formula: see text]，并利用这一事实证明了对于所有 n，[Formula: see text]的黎曼假设。
