Functional equations and uniformity for local zeta functions of nilpotent groups

1996 ◽  
Vol 118 (1) ◽  
pp. 39-90 ◽  
Author(s):  
Marcus Du Sautoy ◽  
Alexander Lubotzky
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Nilpotent Groups ◽  
Local Zeta Functions

scholarly journals IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

2020 ◽  
Vol 71 (3) ◽  
pp. 959-980
Author(s):  
Christopher Voll
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Infinite Family ◽  
Nilpotent Groups ◽  
Unipotent Group ◽  
Group Schemes ◽  
Lie Rings ◽  
Local Functional ◽  
Explicit Formulae ◽  
Local Functional Equations

Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

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, II: Groups of type F, G, and H

2020 ◽  
Vol 30 (05) ◽  
pp. 931-975
Author(s):  
Paula Macedo Lins de Araujo
Keyword(s):  
Conjugacy Class ◽  
Functional Equations ◽  
Zeta Functions ◽  
Nilpotent Groups ◽  
Number Fields ◽  
Unipotent Group ◽  
Group Schemes ◽  
Local Factors ◽  
Joint Distributions ◽  
Rings Of Integers

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class [Formula: see text] generalizing the Heisenberg group of ([Formula: see text])-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups.

Holomorphy of local zeta functions for curves

1993 ◽  
Vol 295 (1) ◽  
pp. 635-641 ◽  
Author(s):  
Willem Veys
Keyword(s):  
Zeta Functions ◽  
Local Zeta Functions

scholarly journals Computing -series of geometrically hyperelliptic curves of genus three

2016 ◽  
Vol 19 (A) ◽  
pp. 220-234 ◽  
Author(s):  
David Harvey ◽  
Maike Massierer ◽  
Andrew V. Sutherland
Keyword(s):  
Quadratic Field ◽  
Algebraic Closure ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
Good Reduction ◽  
Usual Form ◽  
Local Zeta Functions ◽  
Curves Of Genus Three

Let$C/\mathbf{Q}$be a curve of genus three, given as a double cover of a plane conic. Such a curve is hyperelliptic over the algebraic closure of$\mathbf{Q}$, but may not have a hyperelliptic model of the usual form over$\mathbf{Q}$. We describe an algorithm that computes the local zeta functions of$C$at all odd primes of good reduction up to a prescribed bound$N$. The algorithm relies on an adaptation of the ‘accumulating remainder tree’ to matrices with entries in a quadratic field. We report on an implementation and compare its performance to previous algorithms for the ordinary hyperelliptic case.

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