Higher-rank zeta functions and SL _n -zeta functions for curves

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 whenG=SLnand 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 ton=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.

Higher-rank zeta functions for elliptic curves

“…This equality, which was conjectured in ref. 2, is proved in the companion paper (4). In this paper we concentrate on the case of elliptic curves X = E , i.e., g = 1, where we can give much more complete results.…”

“…For the zeta function, on the other hand, it is crucial (for reasons indicated briefly in ref. 4, remark 1) to restrict to the opposite case n|d . The following properties of ζX ,n (s) were shown in ref.…”