Stability results for local zeta functions of groups algebras, and modules

2017 ◽  
Vol 165 (3) ◽  
pp. 435-444 ◽  
Author(s):  
TOBIAS ROSSMANN
Keyword(s):  
Group Theory ◽  
Functional Equations ◽  
Zeta Functions ◽  
Local Zeta Functions ◽  
Single Formula ◽  
Almost All ◽  
Stability Results ◽  
Zeta Functions Of Groups ◽  
Local Base

AbstractVarious types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

scholarly journals Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties

2019 ◽  
Vol 22 (4) ◽  
pp. 741-774 ◽  
Author(s):  
Paula Macedo Lins de Araujo
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Number Fields ◽  
Nilpotency Class ◽  
Unipotent Group ◽  
Group Schemes ◽  
Isomorphism Classes ◽  
Arithmetic Properties ◽  
Rings Of Integers ◽  
Almost All

AbstractThis is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.

scholarly journals Local zeta functions and Newton polyhedra

2003 ◽  
Vol 172 ◽  
pp. 31-58 ◽  
Author(s):  
W. A. Zuniga-Galindo
Keyword(s):  
Zeta Function ◽  
Valuation Ring ◽  
Zeta Functions ◽  
Newton Polyhedron ◽  
Group Of Units ◽  
Local Zeta Function ◽  
Degenerate Polynomial ◽  
Small Set ◽  
Local Zeta Functions ◽  
Almost All

AbstractTo a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

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