Higher-rank zeta functions and SLn-zeta functions for curves

Lin Weng; Don Zagier

doi:10.1073/pnas.1912501117

New Research In

Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved January 15, 2020 (received for review July 19, 2019)

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Significance

Almost 100 years ago, Artin defined an analog of the famous Riemann zeta function for curves (one-dimensional varieties) over a finite field. In 2005, L.W. defined two different series of “higher zeta functions” for curves over finite fields that both generalized Artin’s zeta functions, one being defined geometrically and the other using advanced concepts from group representation theory, and conjectured that they always coincide. In this paper this conjecture is proved by giving a formula for one of the two series and showing that it agrees with the formula for the other series proved a few years ago by Sergey Mozgovoy and Markus Reineke.

Abstract

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when G=SLn and P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to n=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.