scholarly journals Isomorphism Classes and Zeta-functions of Some Nilpotent Groups II

2015 ◽  
Vol 38 (2) ◽  
pp. 523-537
Author(s):  
Fumitake HYODO
Keyword(s):  
Zeta Functions ◽  
Nilpotent Groups ◽  
Isomorphism Classes

scholarly journals Higher-rank zeta functions for elliptic curves

2020 ◽  
Vol 117 (9) ◽  
pp. 4546-4558 ◽  
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Vector Bundles ◽  
Zeta Functions ◽  
Smooth Curve ◽  
Isomorphism Classes ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

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.

scholarly journals Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexander Stasinski ◽  
Christopher Voll
Keyword(s):  
Zeta Functions ◽  
Nilpotent Groups ◽  
Number Fields ◽  
Vector Spaces ◽  
Unipotent Group ◽  
Finitely Generated ◽  
Group Schemes ◽  
Prehomogeneous Vector Spaces ◽  
Rings Of Integers

AbstractWe compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating

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.

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