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Gunther Cornelissen

Publications and source records attributed to Gunther Cornelissen.

At least 19 recordsLinked to original sources
arXiv2026

The asymptotic Mahler measure of Gaussian periods

We construct a sequence of cyclotomic integers (Gaussian periods) of particularly small Mahler measure/height. We study the asymptotics of their Mahler measure as a function of their conductor, to find that the growth rate is the (multivariate) Mahler measure of a family of log Calabi-Yau varieties of increasing dimension. In turn, we study the asymptotics of some of these Mahler measures as the dimension increases, as well as properties of the associated algebraic dynamical system. We describe computational experiments that suggest that these cyclotomic integers realise the smallest non-zero logarithmic Mahler measure in the set of algebraic integers with cyclic Galois group of a given odd order. Finally, we discuss some precise conjectures that imply double logarithmic growth for those Mahler measures as a function of that order. The proofs use ideas from the theory of quantitative equidistribution, reflexive polytopes and toric varieties, the theory of random walks, Bessel functions, class field theory, and Linnik's constant.

math.NT↗
arXiv2026

Orbit decomposition statistics for discrete dynamical systems: the Cesàro mean and a large deviation principle

If a self-map $σ\colon \mathcal{X} \rightarrow \mathcal{X}$ has a dynamical zeta function with nonzero radius of convergence $1/Λ$ and the Cesàro mean $B$ of $ \# \mathrm{Fix}(σ^k)/Λ^k$ exists and is positive, we show a large deviation principle for the number of prime orbits occurring in the decomposition of a general orbit of length $\leq X$ (an element of the free abelian monoid generated by the prime orbits or, equivalently, a prime orbit of a finite multiset in $\mathcal{X}$) with speed $B \log X$ and universal rate function equal to that of the Poisson distribution with unit mean. We also show a large deviation principle for more general strongly additive functions. The proof uses asymptotic results on the total number of general orbits, as well as a weak analogue of Mertens's second theorem, that may be of independent interest. The theory applies, for example, to endomorphisms of algebraic groups over finite fields, additive cellular automata, and automorphisms of some solenoids.

math.DS↗
arXiv2025

Geodesic clustering of zeros of Eisenstein series for congruence groups

We consider a set of generators for the space of Eisenstein series of even weight $k$ for any congruence group $Γ$ and study the set of all of their zeros taken for $Γ(1)$-conjugates of $Γ$ in the standard fundamental domain for $Γ(1)$. We describe (a) an upper bound $κ_Γ+ O(1/k)$ for their imaginary part; (b) a finite configuration of geodesics segments to which all zeros converge in Hausdorff distance as $k \rightarrow \infty$; (c) a finite set containing all algebraic zeros for all weights. The bound in (a) depends on the (non-)vanishing of a new generalization of Ramanujan sums. The proof of (b) originates in a method used to study phase transitions in statistical physics. The proof of (c) relies on the theory of complex multiplication. The results can be made quantitative for specific groups. For $Γ=Γ(N)$ with $4 \nmid N$, $κ_Γ=1$ and the zeros tend to the unit circle, whereas if $4 \mid N$, $κ_Γ=2$ and the limit configuration includes parts of vertical geodesics and circles of radius $2$. In both cases, the only algebraic zeros are at $\mathrm{i}$ and $\exp(2π\mathrm{i}/3)$ for sufficiently large $k$. For $Γ(N)$ with $N$ odd, we use finer estimates to prove a trichotomy for the exact `convergence speed' of the zeros to the unit circle, as well as angular equidistribution of the zeros as $k \rightarrow \infty$.

math.NT↗
arXiv2024

Dynamics of endomorphisms of algebraic groups

Let $σ$ denote an endomorphism of a smooth algebraic group $G$ over the algebraic closure of a finite field, and assume all iterates of $σ$ have finitely many fixed points. Steinberg gave a formula for the number of fixed points of $σ$ (and hence of all of its iterates $σ^n$) in the semisimple case, leading to a representation of its Artin-Mazur zeta function as a rational function. We generalise this to an arbitrary (smooth) algebraic group $G$, where the number of fixed points $σ_n$ of $σ^n$ can depend on $p$-adic properties of $n$. We axiomatise the structure of the sequence $(σ_n)$ via the concept of a `finite-adelically distorted' (FAD-)sequence. Such sequences also occur in topological dynamics, and our subsequent results about zeta functions and asymptotic counting of orbits apply equally well in that situation; for example, to $S$-integer dynamical systems, additive cellular automata and other compact abelian groups. We prove dichotomies for the associated Artin-Mazur zeta function, and study the analogue of the Prime Number Theorem for the function counting periodic orbits of length $\leq N$. For an algebraic group $G$ we express the error term via the $\ell$-adic cohomological zeta function of $G$.

math.NT↗
arXiv2024

Multiband linear cellular automata and endomorphisms of algebraic vector groups

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.

math.DS↗
arXiv2022

Problems hard for treewidth but easy for stable gonality

We show that some natural problems that are XNLP-hard (which implies W[t]-hardness for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds (which is strongly NP-complete, as shown by Itai), Minimum Maximum Outdegree (for which W[1]-hardness for treewidth was proven by Szeider), and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set (for which W[1]-hardness was proven by Dom, Lokshtanov, Saurabh and Villanger). Our hardness proofs (that beat existing results) use reduction to a recent XNLP-complete problem (Accepting Non-deterministic Checking Counter Machine). The new easy parameterized algorithms use a novel notion of weighted tree partition with an associated parameter that we call treebreadth, inspired by Seese's notion of tree-partite graphs, as well as techniques from dynamical programming and integer linear programming.

cs.DS↗
arXiv2021

Twisted isospectrality, homological wideness and isometry

Given a manifold (or, more generally, a developable orbifold) $M_0$ and two closed Riemannian manifolds $M_1$ and $M_2$ with a finite covering map to $M_0$, we give a spectral characterisation of when they are equivalent Riemannian covers (in particular, isometric), assuming a representation-theoretic condition of "homological wideness": if $M$ is a common finite cover of $M_1$ and $M_2$ and $G$ is the covering group of $M$ over $M_0$, the condition involves the action of $G$ on the first homology group of $M$ (it holds, for example, when there exists a rational homology class on $M$ whose orbit under $G$ consists of $|G|$ linearly independent homology classes). We prove that, under this condition, Riemannian covering equivalence is the same as isospectrality of finitely many twisted Laplacians on the manifolds, acting on sections of flat bundles corresponding to specific representations of the fundamental groups of the manifolds involved. Using the same methods, we provide spectral criteria for weak conjugacy and strong isospectrality. In the negative curvature case, we formulate an analogue of our result for the length spectrum. The proofs are inspired by number-theoretical analogues. We study examples where the representation theoretic condition does and does not hold. For example, when $M_1$ and $M_2$ are commensurable non-arithmetic closed Riemann surfaces of negative Euler characteristic, there is always such an $M_0$, and the condition of homological wideness always holds.

math.DG↗
arXiv2020

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest $d\geq 1$ for which $a_2=\dots=a_d=0$. Only a handful of power series of finite order are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series. Up to conjugation, there are only finitely many series $σ$ of order $2^n$ with fixed break sequence (i.e. the sequence of depths of $σ^{\circ 2^i}$). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2. We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth $2^μ\pm 1$ $(μ\geq 1)$. Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states, the minimal possible number for such a series.

math.NT↗
arXiv2019

Recognizing hyperelliptic graphs in polynomial time

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph.

cs.DS↗
arXiv2019

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the moduli space of rank-2 Drinfeld modules with principal level structure preserve the cusps.

math.AG↗
arXiv2019

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.

math.NT↗
arXiv2018

Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.

math.CO↗
arXiv2018

Dynamics on abelian varieties in positive characteristic

We study periodic points for endomorphisms $σ$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $ζ_σ$ of $σ$ is either rational or transcendental, the first case happening precisely when $σ^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $σ$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_σ$, in which the number of fixed points of $σ^n$ is replaced by the degree of $σ^n-1$, is always a rational function. Let $1/Λ$ denote its largest real pole and assume no other pole or zero has the same absolute value. Then, using a general dichotomy result for power series proven by Royals and Ward in the appendix, we find that $ζ_σ(z)$ has a natural boundary at $|z|=1/Λ$ when $σ$ is not very inseparable. We introduce and study tame dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $ζ^*_σ$ that is always algebraic, and such that $ζ_σ$ factors into an infinite product of tame zeta functions. We briefly discuss functional equations. Finally, we study the length distribution of orbits and tame orbits. Orbits of very inseparable endomorphisms distribute like those of Axiom A systems with entropy $\log Λ$, but the orbit length distribution of not very inseparable endomorphisms is more erratic and similar to $S$-integer dynamical systems. We provide an expression for the prime orbit counting function in which the error term displays a power saving depending on the largest real part of a zero of $D_σ(Λ^{-s})$.

math.NT↗
arXiv2016

Rigidity and reconstruction for graphs

We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths".

math.CO↗
arXiv2015

Convergence of families of Dirichlet series

We give some conditions under which (uniform) convergence of a family of Dirichlet series to another Dirichlet series implies the convergence of their individual coefficients and/or exponents. We give some applications to some spectral zeta functions that arise in Riemannian geometry and physics.

math.CA↗
arXiv2015

Hecke algebra isomorphisms and adelic points on algebraic groups

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and $G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$ that are Galois over $\mathbf{Q}$, the finite Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n > 1$) are isomorphic by an isometry for the $L^1$-norm, then the fields $K$ and $L$ are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over $\mathbf{Q}$.

math.NT↗