Publications

Journal articles

1. Irreducibility testing of finite nilpotent linear groups. J. Algebra 324 (2010), 1114–1124.
   DOI preprint
   abstract

   We describe an algorithm for irreducibility testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For a reducible group, our algorithm constructs a proper submodule. An implementation in Magma is publicly available.

2. Primitivity testing of finite nilpotent linear groups. LMS J. Comput. Math. 14 (2011), 87–98.
   DOI preprint
   abstract

   We describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

3. Computing topological zeta functions of groups, algebras, and modules, I. Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1099–1134.
   DOI arXiv:1405.5711 preprint
   abstract

   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.

4. Computing topological zeta functions of groups, algebras, and modules, II. J. Algebra 444 (2015), 567–605.
   DOI arXiv:1409.5044 preprint
   abstract

   Building on our previous work, we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, possibly non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.

5. Topological representation zeta functions of unipotent groups. J. Algebra 448 (2016), 210–237.
   DOI arXiv:1503.01942 preprint
   abstract

   Inspired by work surrounding Igusa’s local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established \(p\)-adic representation zeta functions associated with pro-\(p\) groups derived from unipotent groups. We investigate fundamental properties of the topological zeta functions considered here. We also develop a method for computing them under non-degeneracy assumptions. As an application, among other things, we obtain a complete classification of topological representation zeta functions of unipotent algebraic groups of dimension at most six.

6. Primitive finite nilpotent linear groups over number fields. J. Algebra 451 (2016), 248–267.
   DOI arXiv:1506.00811 preprint
   abstract

   Building upon the author’s previous work on primitivity testing of finite nilpotent linear groups over fields of characteristic zero, we describe precisely those finite nilpotent groups which arise as primitive linear groups over a given number field. Our description is based on arithmetic conditions involving invariants of the field.

7. Enumerating submodules invariant under an endomorphism. Math. Ann. 368 (2017), no. 1, 391–417.
   DOI arXiv:1606.00760 preprint
   abstract

   We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of (\(S\)-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence \(2\).

8. Computing local zeta functions of groups, algebras, and modules. Trans. Amer. Math. Soc. 370 (2018), no. 7, 4841–4879.
   DOI arXiv:1602.00919 preprint
   abstract

   We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with \(\mathfrak{gl}_2(\mathbf Q)\). Finally, we introduce and compute examples of graded subobject zeta functions.

9. The average size of the kernel of a matrix and orbits of linear groups. Proc. Lond. Math. Soc. (3) 117 (2018), no. 3, 574–616.
   DOI arXiv:1704.02668 preprint
   abstract

   Let \(\mathfrak{O}\) be a compact discrete valuation ring of characteristic zero. Given a module \(M\) of matrices over \(\mathfrak{O}\), we study the generating function encoding the average sizes of the kernels of the elements of \(M\) over finite quotients of \(\mathfrak{O}\). We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules \(M\). Using \(p\)-adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro-\(p\) groups.

10. Stability results for local zeta functions of groups, algebras, and modules. Math. Proc. Camb. Phil. Soc. 165 (2018), no. 3, 435–444.
    DOI arXiv:1504.04164 preprint
    abstract

    Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density \(1\) in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

11. The average size of the kernel of a matrix and orbits of linear groups, II: duality. J. Pure Appl. Algebra 224 (2020), no. 4, 106203, 28 pages.
    DOI arXiv:1807.01101 preprint
    abstract

    Define a module representation to be a linear parameterisation of a collection of module homomorphisms over a ring. Generalising work of Knuth, we define duality functors indexed by the elements of the symmetric group of degree three between categories of module representations. We show that these functors have tame effects on average sizes of kernels. This provides a general framework for and a generalisation of duality phenomena previously observed in work of O’Brien and Voll and in the predecessor of the present article. We discuss applications to class numbers and conjugacy class zeta functions of \(p\)-groups and unipotent group schemes, respectively.

12. Enumerating conjugacy classes of graphical groups over finite fields. Bull. Lond. Math. Soc. 54 (2022), no. 5, 1923–1943.
    DOI arXiv:2107.05564 preprint
    abstract

    Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to their sizes.

13. Linear relations with disjoint supports and average sizes of kernels (with Angela Carnevale). J. Lond. Math. Soc. 106 (2022), no. 3, 1759–1809.
    DOI arXiv:2009.00937 preprint
    abstract

    We study the effects of imposing linear relations within modules of matrices on average sizes of kernels. The relations that we consider can be described combinatorially in terms of partial colourings of grids. The cells of these grids correspond to positions in matrices and each defining relation involves all cells of a given colour. We prove that imposing such relations arising from “admissible” partial colourings has no effect on average sizes of kernels over finite quotients of discrete valuation rings. This vastly generalises the known fact that average sizes of kernels of general square and traceless matrices of the same size coincide over such rings. As a group-theoretic application, we explicitly determine zeta functions enumerating conjugacy classes of finite \(p\)-groups derived from free class-\(3\)-nilpotent groups for \(p\ge 5\).

14. Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraints (with Christopher Voll). Mem. Amer. Math. Soc. 294 (2024), no. 1465, vi+120 pp.
    DOI arXiv:1908.09589 preprint
    abstract

    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.

15. On the enumeration of orbits of unipotent groups over finite fields. Proc. Amer. Math. Soc. 153 (2025), no. 2, 479–495.
    DOI arXiv:2208.04646 preprint
    abstract

    We show that the enumeration of linear orbits and conjugacy classes of \(\mathbf{Z}\)-defined unipotent groups over finite fields is “wild” in the following sense: given an arbitrary scheme \(Y\) of finite type over \(\mathbf{Z}\) and integer \(n\ge 1\), the numbers \(\# Y(\mathbf{F}_q) \bmod q^n\) can be expressed, uniformly in \(q\), in terms of the numbers of linear orbits (or numbers of conjugacy classes) of finitely many \(\mathbf{Z}\)-defined unipotent groups over \(\mathbf{F}_q\) and finitely many Laurent polynomials in \(q\).

16. Coloured shuffle compatibility, Hadamard products, and ask zeta functions (with Angela Carnevale and Vassilis Dionyssis Moustakas). Bull. Lond. Math. Soc. 57 (2025), no. 7, 2132–2154.
    DOI arXiv:2407.01387 preprint
    abstract

    We devise an explicit method for computing combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so-called ask zeta functions of direct sums of modules of matrices or class- and orbit-counting zeta functions of direct products of nilpotent groups. Our method relies on shuffle compatibility of coloured permutation statistics and coloured quasisymmetric functions, extending recent work of Gessel and Zhuang.

Book chapters and conference proceedings

17. Periodicities for graphs of \(p\)-groups beyond coclass (with Bettina Eick). In: Kappe et al. (eds), Computational group theory and the theory of groups, II. Contemp. Math. 511 (2010), 11–23.
    DOI preprint
    abstract

    We use computational methods to investigate periodic patterns in the graphs \(\mathcal{G}(p,(d,w,o))\) associated with the \(p\)-groups of rank \(d\), width \(w\), and obliquity \(o\). In the smallest interesting case \(\mathcal{G}(p,(3,2,0))\) we obtain a conjectural description of this graph for all \(p \geq 3\); in particular, we conjecture that this graph is virtually periodic for all \(p \geq 3\). We also investigate other related infinite graphs.

18. A Framework for Computing Zeta Functions of Groups, Algebras, and Modules. In: Böckle et al. (eds), Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory (2017), Springer-Verlag, 561–586..
    DOI preprint
    abstract

    We give an overview of the author’s recent work on methods for explicitly computing various types of zeta functions associated with algebraic counting problems. Among the types of zeta functions that we consider are the so-called topological ones.

19. From coloured permutations to Hadamard products and zeta functions (with Angela Carnevale and Vassilis Dionyssis Moustakas). Proceedings of the 36th Conference on Formal Power Series and Algebraic Combinatorics (Bochum). Séminaire Lotharingien de Combinatoire 91B (2024). Article #56, 12 pp.
    URL preprint
    abstract

    We devise a constructive method for computing explicit combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so-called ask zeta functions of direct sums of modules of matrices or class- and orbit-counting zeta functions of direct products of groups. Our method relies on shuffle compatibility of coloured permutation statistics and coloured quasisymmetric functions, extending recent work of Gessel and Zhuang.

Preprints

20. Ask zeta functions of joins of graphs (with Christopher Voll). 60 pages.
    arXiv:2505.10263 preprint
    abstract

    In previous work, 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 \(W_{\Gamma}^{\sharp}(X,T)\), 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.

Reports

1. Computing with nilpotent linear groups. Oberwolfach Reports 8 (2011), no. 3, Computational Group Theory, 2121.
   DOI preprint
2. Computing zeta functions of groups, algebras, and modules. Oberwolfach Reports 13 (2016), no. 3, Computational Group Theory, 2144–2145.
   DOI preprint
3. Towards a symbolic enumeration of orbits. Oberwolfach Reports 38/2021, Computational Group Theory, 2064–2066.
   DOI preprint
4. Orbits: tame and wild. Oberwolfach Reports 27/2025, Computational Group Theory.
   DOI preprint
5. Reduced and topological zeta functions in enumerative algebra. Oberwolfach Reports 38/2025, Cohomology Theories for Automorphic Forms and Enumerative Algebra.
   DOI preprint