arXiv subjects
Publications and source records attributed to Christopher Voll.
We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by two distinct invariants, of sublattices of an $\mathfrak{o}$-lattice of finite rank endowed with a non-degenerate symplectic form.
We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary for their meromorphic continuation beyond their abscissa of convergence. We analyse both the number field and function field case.
For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about the arithmetic functions averaging $\ell$th Ehrhart coefficients.
In previous work (arXiv:1908.09589), we studied rational generating functions ("ask zeta functions") associated with graphs and hypergraphs. These functions encode average sizes of kernels of generic matrices with support constraints determined by the graph or hypergraph in question, with applications to the enumeration of linear orbits and conjugacy classes of unipotent groups. In the present article, we turn to the effect of a natural graph-theoretic operation on associated ask zeta functions. Specifically, we show that two instances of rational functions, $W^-_\Gamma(X,T)$ and $W^\sharp_\Gamma(X,T)$, associated with a graph $\Gamma$ are both well-behaved under taking joins of graphs. In the former case, this has applications to zeta functions enumerating conjugacy classes associated with so-called graphical groups.
We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials.
Given a lattice polytope $P$ and a prime $p$, we define a function from the set of primitive symplectic $p$-adic lattices to the rationals that extracts the $\ell$th coefficient of the Ehrhart polynomial of $P$ relative to the given lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type $\mathsf{A}$, we show that these functions are eigenfunctions of a suitably defined action of the spherical symplectic Hecke algebra. Although they depend significantly on the polytope $P$, their eigenvalues are independent of $P$ and expressed as polynomials in $p$. We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat--Tits buildings associated with $p$-adic symplectic groups. We compute these zeta functions by enumerating $p$-adic lattices by their elementary divisors and, simultaneously, one Hermite parameter. We report on a general functional equation satisfied by these local zeta functions, confirming a conjecture of Vankov.
We describe an effective procedure to compute the local subgroup zeta functions of the free class-$2$-nilpotent groups on $d$ generators, for all $d$. For $d=4$, this yields a new, explicit formula. For $d\in\{4,5\}$, we compute the topological subgroup zeta function. We also obtain general results about the reduced and topological subalgebra zeta functions. For the former, we determine the behaviour at one and for the latter, the degree and behaviours at zero and infinity. Some of these results confirm, in the relevant special cases, general conjectures by T. Rossmann.
We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincar\'e series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincar\'e series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert-Poincar\'e series, which we call the coarse flag Hilbert-Poincar\'e series, exhibits intriguing nonnegativity features and - in the case of Coxeter arrangements - connections with Eulerian polynomials. For numerous classes and examples of hyperplane arrangements, we determine their (coarse) flag Hilbert-Poincar\'e series. Some computations were aided by a SageMath package we developed.
We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be sufficient for local functional equations of the generic Euler factors of these zeta functions. This generalizes and unifies previous work on submodule zeta functions including, specifically, ideal zeta functions of nilpotent (Lie) rings and their graded analogues.
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 [1], 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.
We describe the automorphism groups of finite $p$-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of $j$-invariant $1728$ given in Weierstrass form. We interpret these orders in terms of the numbers of $3$-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.
We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of $\mathbf{F}_q$-points of the groups under consideration depend polynomially on $q$. Our approach combines group theory, graph theory, toric geometry, and $p$-adic integration. Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.
We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.
We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed rank in terms of statistics on the hyperoctahedral groups.
We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz's $q$-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series' abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings $\mathfrak{f}_{c,d}$ of nilpotency class $c$ for all $c\leq2$ and for $(c,d)\in\{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.
We 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 $\mathfrak{p}$-adic integrals associated to certain rank varieties of matrices of linear forms.