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 [Formula: see text] 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 [Formula: see text]) 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.
journal_name
Proc Natl Acad Sci U S Aauthors
Weng L,Zagier Ddoi
10.1073/pnas.1912501117subject
Has Abstractpub_date
2020-03-24 00:00:00pages
6398-6408issue
12eissn
0027-8424issn
1091-6490pii
1912501117journal_volume
117pub_type
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