HIGHER RANK ZETA FUNCTIONS FOR ELLIPTIC CURVES
HIGHER RANK ZETA FUNCTIONS FOR ELLIPTIC CURVES
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椭圆曲线的高阶 Zeta 函数
DOI：
10.1073/pnas.1912023117
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发表时间：
2020
影响因子：
11.1
通讯作者：
Don Zagier
中科院分区：
文献类型：
--
作者：
Lin WENG;Don Zagier
在LW的早期工作中，对有限域Fq上的任意光滑曲线X和任意整数n≥ 1定义了一个非交换zeta函数，其中和是在X上秩为n且度可被n整除的Fq-有理半稳定向量丛V的同构类上.这个函数，与X/F q的通常的Artin zeta函数（如果n= 1）一致，是q− s的有理数函数，分母为（1− q− ns）（1− qn− ns），并且在理论上满足黎曼假设。本文详细研究了亏格为1的曲线的情形。我们证明了在这种情况下，Dirichlet级数的和现在是在X上的0次Fq-有理半稳定向量丛V的同构类上，等于<$k= 1∞ <$X/Fq（s+ k），并利用这一事实证明了对所有n的<$X，n（s）的黎曼假设。
In earlier work by LW, 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− 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 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.