scholarly journals Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms

2017 ◽  
Vol 2019 (7) ◽  
pp. 2137-2176 ◽  
Author(s):  
Christopher Voll
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Local Functional ◽  
Nilpotent Algebras ◽  
Local Functional Equations

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 Zeta functions of the 3-dimensional almost-Bieberbach groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego Sulca
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Euler Product ◽  
3 Dimensional ◽  
Local Factors ◽  
Local Functional ◽  
Finite Sums ◽  
Complete Computation ◽  
Local Functional Equations ◽  
Torsion Free

Abstract The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

scholarly journals A nilpotent group without local functional equations for pro-isomorphic subgroups

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Mark N. Berman ◽  
Benjamin Klopsch
Keyword(s):  
Zeta Function ◽  
Nilpotent Group ◽  
Functional Equations ◽  
Finitely Generated ◽  
Local Functional ◽  
Local Functional Equations ◽  
Torsion Free

AbstractThe pro-isomorphic zeta function ζWe manufacture the first example of a torsion-free finitely generated nilpotent group Γ such that the local Euler factors ζ

scholarly journals Functional equations of Weng's zeta functions for (G,P)/Q

2013 ◽  
Vol 135 (4) ◽  
pp. 1019-1038 ◽  
Author(s):  
Yasushi Komori
Keyword(s):  
Functional Equations ◽  
Zeta Functions
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