TY - JOUR

T1 - Higher-rank zeta functions for elliptic curves

AU - Weng, Lin

AU - Zagier, Don

N1 - Funding Information:
ACKNOWLEDGMENTS. L.W. thanks the Japan Society for the Promotion of Science, which partially supported this work. We also thank the Max Planck Institute for Mathematics and Kyushu University for providing excellent research environments.
Publisher Copyright:
© 2020 National Academy of Sciences. All rights reserved.

PY - 2020/3/3

Y1 - 2020/3/3

N2 - 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}| 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 − 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 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 ζX/Fq(s + k), and use this fact to prove the Riemann hypothesis for ζX ,n(s) for all n.

AB - 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}| 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 − 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 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 ζX/Fq(s + k), and use this fact to prove the Riemann hypothesis for ζX ,n(s) for all n.

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U2 - 10.1073/pnas.1912023117

DO - 10.1073/pnas.1912023117

M3 - Article

C2 - 32071252

AN - SCOPUS:85081140528

SN - 0027-8424

VL - 117

SP - 4546

EP - 4558

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 9

ER -