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Journal Article

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


Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Weng, L., & Zagier, D. (2020). Higher-rank zeta functions and SLn-zeta functions for curves. Proceedings of the National Academy of Sciences of the United States of America, 117(12), 6398-6408. doi:10.1073/pnas.1912501117.

Cite as: https://hdl.handle.net/21.11116/0000-0006-3AB5-D
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.