Christopher Voll

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

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

A

Normal zeta functions of the Heisenberg groups over number rings I: the unramified case

. 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.

COUNTING SUBGROUPS IN A FAMILY OF NILPOTENT SEMI-DIRECT PRODUCTS

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adelic profinite groups of type A2. This has consequences for the representation zeta functions of arithmetic groups Γ⊂H(k), where k is a number field and H is a k-form of SL3: 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 SL3(o) and SU3(o), where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3×3 matrices in gl3(o) and gu3(o), where 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 SL3(o), SU3(o), GL3(o), and GU3(o), arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of ‘Ennola duality’.

Lectures on Profinite Topics in Group Theory: An introduction to compact p -adic Lie groups

Let

Orbit Dirichlet series and multiset permutations

K

Enumerating graded ideals in graded rings associated to free nilpotent Lie rings

be a number field with ring of integers

Enumerating traceless matrices over compact discrete valuation rings

\mathcal{O}_K

Functional equations for zeta functions of groups and rings

. We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups

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

H(\mathcal{O}_K)