Higher-rank zeta functions for elliptic curves

Lin Weng, Don Zagier

Research output: Contribution to journalArticle

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 ζX/Fq,n(s) = X |H0(X, V)r{0}| qdeg(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 − qnns) 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)| qrank(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 ζX/Fq(s + k), and use this fact to prove the Riemann hypothesis for ζX ,n(s) for all n.

Original languageEnglish
Pages (from-to)4546-4558
Number of pages13
JournalProceedings of the National Academy of Sciences of the United States of America
Volume117
Issue number9
DOIs
Publication statusPublished - Mar 3 2020

All Science Journal Classification (ASJC) codes

  • General

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