Zeta functions of three-dimensional p-adic Lie algebras

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or --)groups, and the normal zeta functions of --groups of class 2. We deduce our theorems from a "blueprint result" on certain p-adic integrals which generalises work of Denef and others on Igusa's local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to "linearise" the problems of counting subgroups and representations in --groups, respectively.

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“…In [22] we gave a formula for the zeta function of an arbitrary 3-dimensional ‫ޚ‬ p -Lie algebra, based on the proof of Theorem A.…”

Section: Introductionmentioning

confidence: 99%

Functional equations for zeta functions of groups and rings

Voll

2010

Ann. Math.

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123

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“…This geometrically minded computation was later reinterpreted in a motivic setting by du Sautoy and Loeser, allowing them to deduce that ζ sl2(Z),top (s) = 3s − 1 2(2s − 1)(s − 1) 2 s , (7.2) see [21, § 9.3] and note that we already corrected the shift present in [21], see Remark 5.18. For yet another p-adic approach, Klopsch and Voll [35] gave an explicit formula for ζ L (s), where L is a Lie Z p -algebra which is free of rank 3 as a Z p -module, in terms of Igusa's local zeta function associated with a ternary quadratic form attached to L. Setting up the integral. We now explain how the p-adic integral in [23] can, for odd p, be computed using our method.…”

Section: Applicationsmentioning

confidence: 99%

Computing topological zeta functions of groups, algebras, and modules, I

Rossmann

2015

Proceedings of the London Mathematical Society

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We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa‐type zeta functions. At the heart of our method lies an explicit convex‐geometric formula for a class of p‐adic integrals under non‐degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

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“…Numerous other works study the poles of the local zeta function [25,34,20,35,3,36,4,7,5,44,45,30,47,38,39,23,24]. Many authors have labored on the calculation of local zeta functions in various situations [28,37,1,5,42,43,21,29,30,9], and many works either use local zeta functions or else apply the methods developed for obtaining them [8,46,48,31,17,18,49,40,33,50]. Nonetheless, certain classes of polynomials have proved forbidding to those who wish to obtain general results.…”

Section: Introductionmentioning

confidence: 99%

A new generating function for calculating the Igusa local zeta function

Cowan

Katz

White

2017

Advances in Mathematics

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A new method is devised for calculating the Igusa local zeta function Z f of a polynomial f (x1, . . . , xn) over a p-adic field. This involves a new kind of generating function G f that is the projective limit of a family of generating functions, and contains more data than Z f . This G f resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly. Local zeta functions for arbitrary quadratic polynomials over p-adic fields with p odd are presented, as well as for polynomials over unramified 2-adic fields of the form Q + L where Q is a quadratic form and L is a linear form where Q and L have disjoint variables. For a quadratic form over an arbitrary p-adic field with odd p, this new technique makes clear precisely which of the three candidate poles are actual poles.

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Zeta functions of three-dimensional p-adic Lie algebras

Cited by 14 publications

References 18 publications

Functional equations for zeta functions of groups and rings

Functional equations for zeta functions of groups and rings

Computing topological zeta functions of groups, algebras, and modules, I

A new generating function for calculating the Igusa local zeta function

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