Searcharxiv⌕ Search

arXiv subjects

Christopher Voll

Publications and source records attributed to Christopher Voll.

At least 37 records · Page 2Linked to original sources
arXiv2016

Arithmetic groups, base change, and representation growth

Consider an arithmetic group $\mathbf{G}(O_S)$, where $\mathbf{G}$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of $S$-integers $O_S$ of a number field $K$ with respect to a finite set of places $S$. For each $n \in \mathbb{N}$, let $R_n(\mathbf{G}(O_S))$ denote the number of irreducible complex representations of $\mathbf{G}(O_S)$ of dimension at most $n$. The degree of representation growth $α(\mathbf{G}(O_S)) = \lim_{n \rightarrow \infty} \log R_n(\mathbf{G}(O_S)) / \log n$ is finite if and only if $\mathbf{G}(O_S)$ has the weak Congruence Subgroup Property. We establish that for every $\mathbf{G}(O_S)$ with the weak Congruence Subgroup Property the invariant $α(\mathbf{G}(O_S))$ is already determined by the absolute root system of $\mathbf{G}$. To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions $K \subset L$. We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky's conjecture to Serre's conjecture on the weak Congruence Subgroup Property, which it refines.

math.GR↗
arXiv2015

Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $ζ_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $ζ_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf{G}$, independent of $\mathcal{O}$.

math.GR↗
arXiv2015

Similarity classes of integral $p$-adic matrices and representation zeta functions of groups of type $A_2$

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta functions of arithmetic groups $Γ\subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form of $\mathsf{SL}_3$: assuming that $Γ$ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of $Γ$. Our results are based on explicit, uniform formulae for the representation zeta functions of the $p$-adic analytic groups $\mathsf{SL}_3(\mathfrak{o})$ and $\mathsf{SU}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$. These formulae build on our classification of similarity classes of integral $\mathfrak{p}$-adic $3\times3$ matrices in $\mathfrak{gl}_3(\mathfrak{o})$ and $\mathfrak{gu}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form $\mathsf{SL}_3(\mathfrak{o})$, $\mathsf{SU}_3(\mathfrak{o})$, $\mathsf{GL}_3(\mathfrak{o})$, and $\mathsf{GU}_3(\mathfrak{o})$, arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of $\mathfrak{o}$ is either $0$ or sufficiently large. Analysis of some of these formulae leads us to observe $p$-adic analogues of `Ennola duality'.

math.GR↗
arXiv2014

Zeta functions of groups and rings - recent developments

I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations.

math.GR↗
arXiv2014

Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.

math.GR↗
arXiv2014

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-2-nilpotent groups obtained from three infinite families of group schemes, generalising the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type B. This allows us to establish a formula for the joint distribution of three functions, or 'statistics', on such Weyl groups. Finally, we compare our explicit formulae to p-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.

math.GR↗
arXiv2013

A new statistic on the hyperoctahedral groups

We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable "supporting sets" for the relevant generating functions. We prove cancellations on the complements of these supporting sets using suitably defined sign reversing involutions.

math.CO↗
arXiv2010

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, `perfect' Lie lattice satisfy functional equations. In the case of `semisimple' compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by centraliser dimension. Based on this algebro-geometric description, we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups SL_3(O), where O is a compact discrete valuation ring of characteristic 0, and of the corresponding unitary groups. These formulae, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A_2. Assuming a conjecture of Serre on the Congruence Subgroup Problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A_2 defined over number fields.

math.GR↗
arXiv2010

Representation zeta functions of some compact p-adic analytic groups

Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give general estimates for the abscissae of convergence of such zeta functions. We compute explicit formulae for the representation zeta functions of some compact p-adic analytic groups, defined over a compact discrete valuation ring O of characteristic 0. These include principal congruence subgroups of SL_2(O), without any restrictions on the residue field characteristic of O, as well as the norm one group SL_1(D) of a non-split quaternion algebra D over the field of fractions of O and its principal congruence subgroups. We also determine the representation zeta functions of principal congruence subgroups of SL_3(O) in the case that O has residue field characteristic 3 and is unramified over Z_3.

math.GR↗
arXiv2010

On representation zeta functions of groups and a conjecture of Larsen and Lubotzky

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A_2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

math.GR↗
arXiv2008

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

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.

math.GR↗
arXiv2008

Functional equations for zeta functions of groups and rings

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 $\T$-)groups, and the normal zeta functions of $\T$-groups of class 2. In particular we solve the two problems posed in \cite[Section 5]{duSG/06}. 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 $\T$-groups, respectively.

math.GR↗
arXiv2007

Zeta functions of 3-dimensional p-adic Lie algebras

We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The formula for the zeta function is given in terms of Igusa's local zeta function associated to this form.

math.GR↗
arXiv2004

Normal subgroup growth in free class-2-nilpotent groups

Let $F_{2,d}$ denote the free class-2-nilpotent group on $d$ generators. We compute the normal zeta functions $ζ^\triangleleft_{F_{2,d}}(s)$, prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.

math.GR↗