The meromorphic continuation of a zeta function of Weil and Igusa Type

Abstract

In this paper we consider a local zeta function in two complex variables associated to an affine algebraic variety defined over a finite algebraic extension of Qp" The zeta function combines the Well and Igusa zeta functions for this variety in a natural way. We investigate the properties of this function and in particular show that it has a meromorphic continuation to C 2.Let K 1 denote a fixed finite algebraic extension of Qp. Let The numbers N1, e are those that appear in the Weil zeta function. The coefficient of w e is Pc(z)= Y~ S~,eq-~ z e e>O which is the Igusa zeta function for the field K e which was shown to be rational in [6] and [7].

Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero

“…With this notation, the following comparison result between the tower of local zeta functions Z ′ d (T ) for d ≥ 1 and its motivic counterpart Z mot (T ) generalize results of [23] and [26].…”

“…Let us start with some motivation. Let K be a fixed finite field extension of Q p with residue field F q and let K d denote its unique unramified extension of degree d, for d Meuser in [23] proved that there exist polynomials G and H in Z[T, X 1 , • • • , X t ] and complex numbers λ 1 , • • • , λ t such that, for all d ≥ 1, Z d (s) = G(q −ds , q dλ 1 , • • • , q dλt ) H(q −ds , q dλ 1 , • • • , q dλt ) .…”

The Igusa Local Zeta Function Associated with the Singular Cases of the Determinant and the Pfaffian

“…For example, in the Archimedean case the real poles of | f | s (8) are known to be the zeros of the Bernstein polynomial [2] and hence by Malgrange [14] related to an eigenvalue of the local monodromy of f. Igusa has conjectured a similar relationship in the p-adic case [12]. For an excellent survey of the conjectures and results surrounding the Igusa local zeta function, please see Denef's report [4] and the work of Meuser [15,16,17]. Motivated by the need to have a better understanding of the p-adic case, Igusa has determined the local zeta function Z(t)= | f | s (8) for a large number of group invariants f (x), where 8=, X 0 is the characteristic function of the lattice of integral points of K n .…”

The classification of spinors under 𝐺𝑆𝑝𝑖𝑛₁₄ over finite fields

“…We should also mention that the poles of Z(s) for curves have been closely examined by D. Meuser [18,19]. …”