A Functional Equation of Igusa's Local Zeta Function

2010

Ann. Math.

119

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.

Section: Introductionmentioning

confidence: 87%

Functional equations for zeta functions of groups and rings

2010

Ann. Math.

119

“…The main application of the approach developed in [20] is to prove that, given a torsion-free ring L (not necessarily associative or Lie), the associated local zeta function ζ Z p ⊗ Z L (s) satisfies a functional equation for almost all primes p. The occurrence of functional equations for the zeta functions of some three-dimensional Z p -Lie algebras is therefore only explained by the results of [20] in case these algebras are the 'generic' completions of an algebra over Z (such as, for example, sl 2 (Z p ) for odd p). This corresponds to the fact that the proof of a functional equation for Igusa's local zeta function given in [2] critically depends on good reduction modulo p. 4. The case of three-dimensional Lie algebras is the first non-trivial one as far as subalgebra zeta functions are concerned.…”

Section: Lie Algebramentioning

confidence: 99%

“…(2)) and then to use the identities (4) and (3). In fact, the series P f (t) has only two non-zero summands: one easily computes N 0 = 1 and N 1 = p − 1.…”

Section: The 'Simple' Lie Algebra Sl 2 (Z P )mentioning

confidence: 99%

Zeta functions of three-dimensional p-adic Lie algebras

Klopsch

2008

Math. Z.

“…More recently, Stanley established similar symmetries for the Hilbert-Poincaré series of graded algebras, with remarkable applications to counting problems in combinatorics and topology; see [17]. The symmetries of the Weil zeta functions lie at the heart of Denef and Meuser's proof of a functional equation for certain p-adic integrals, called Igusa's local zeta functions; see [3]. The phenomenon of functional equations also arises in the context of zeta functions associated to groups and rings.…”

Section: General Introductionmentioning

confidence: 98%

Igusa-type functions associated to finite formed spaces and their functional equations

Klopsch

Trans. Amer. Math. Soc.

2009

Abstract. We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, Hermitian or quadratic form. To this end we introduce Igusa-type rational functions encoding these polynomials and prove that they satisfy certain functional equations.Some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on Coxeter groups of type A. While our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements, we formulate a precise conjecture how in this situation, too, the polynomials may be described in terms of parabolic length functions.