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.

# Research Publications

Below is a list of my scientific work with abstracts. The titles link to a pdf file of the papers, mostly the arxiv-versions. At the end of the abstract, you can sometimes find remarks about the paper, including corrections of mistakes that I am currently aware of. If you are only interested in my list of publications, see my cv.

### Dynamically affine maps in positive characteristic

(with Jakub Byszewski and Marc Houben), preprint 31 pp. (2019).

### Reconstructing global fields from dynamics in the abelianized Galois group

(with Xin Li, Matilde Marcolli, Harry Smit),
Selecta Math. **25**:24, 18 pp. (2019).

We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry.

### Rigidity and reconstruction for graphs

(with Janne Kool) preprint (2016), 9pp, to appear in J. Fractal Geom.

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".

### Characterization of global fields by Dirichlet L-series

(with Bart de Smit, Xin Li, Matilde Marcolli, Harry Smit), Res. Number Theory 5:7 (2019).

We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.

### Dynamics on abelian varieties in positive characteristic

(with Jakub Byszewski; appendix by Robert Royals and Thomas Ward), Alg. & Numb. Th. **12**, no. 9 (2018), 2185-2235.

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or transcendental, the first case happening precisely when $\sigma^n-1$ is a separable isogeny for all $n$. We call this condition \emph{very inseparability} and show it is equivalent to the action of $\sigma$ on the local $p$-torsion group scheme being nilpotent. The ``false'' zeta function $D_\sigma$, in which the number of fixed points of $\sigma^n$ is replaced by the degree of $\sigma^n-1$, is always a rational function. Let $1/\Lambda$ 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 $\zeta_\sigma(z)$ has a natural boundary at $|z|=1/\Lambda$ when $\sigma$ is not very inseparable. We introduce and study \emph{tame} dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $\zeta^*_{\sigma}$ that is always algebraic, and such that $\zeta_\sigma$ 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 $\log \Lambda$, 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 $D_\sigma(\Lambda^{-s})$.

### Edge reconstruction of the Ihara zeta function

(with Janne Kool)
Electron. J. Combin. **25**, Issue 2, Paper \#P2.26 (22pp., 2018).

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 Bass-Hashimoto edge adjancency operator $T$: it is symmetric on a Krein space 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.

### Recognizing hyperelliptic graphs in polynomial time (with J. Bodewes, H.L. Bodlaender and M. van der Wegen)

Proc. 44th International Workshop Graph-Theoretic Concepts in CS (WG 2018), Springer LNCS vol. 11159, pp. 52-64 (2018).

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.

**Remark.** Won the Best Student Paper award at WG 2018.

### Hecke algebra isomorphisms and adelic points on algebraic groups

(with Valentijn Karemaker)
Doc. Math. **22** (2017), 851–871.

Let $G$ denote an algebraic group over $\mathbf Q$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism $G(\mathbf{A}_K) \cong G(\mathbf{A}_L)$ of points on $G$ over the adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure and the splitting field 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 $G(\mathbf{A}_K) \cong G(\mathbf{A}_L)$, then $K$ and $L$ are isomorphic as fields.
As a corollary, we show that an isomorphism of Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n \geq 2$), which is an isometry in the $L^1$-norm over two number fields $K$ and $L$ that are Galois over $\mathbf Q$, implies that 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 $\mathbf Q$.

### Distances in spaces of physical models: partition functions versus spectra

(with Aristides Kontogeorgis), Lett. Math. Phys. **107**, Issue 1 (2017), 129-144.

WWe study the relation between convergence of partition functions (seen as general Dirich- let series) and convergence of spectra and their multiplicities. We describe applications to convergence in physical models, e.g., related to topology change and averaging in cosmology.

### The perfect power problem for elliptic curves over function fields

(with Jonathan Reynolds)
New York J. Math. **22** (2016), 95-114.

We generalise the Siegel-Voloch theorem about $S$-integral points on elliptic curves as follows: let $K/F$ denote a global function field over a finite field $F$ of characteristic $p \geq 5$, let $S$ denote a finite set of places of $K$ and let $E/K$ denote an elliptic curve over $K$ with $j$-invariant $j_E \notin K^p$. Fix a function $f \in K(E)$ with a pole of order $N>0$ at the zero of $E$. We prove that there are only finitely many rational points $P \in E(K)$ such that for any valuation outside $S$ for which $f(P)$ is negative, that valuation of $f(P)$ is divisible by some integer not dividing $N$. We also present some effective bounds for certain elliptic curves over rational function fields.

### Reconstruction problems in number theory in the light of C*-algebras

Oberwolfach Report **2016/6** (2016), 20–22.

A survey paper based on a talk at a mini-workshop on Operator Spaces.

### A combinatorial Li-Yau inequality and rational points on curves

(with Fumiharu Kato and Janne Kool), Math. Ann. **361**, Issue 1, pp. 211-258 (2015).

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $k$ denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of $k$ in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some "volume" of the original graph; this can be seen as a substitute for graphs of the Li-Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups $\Gamma$ of $\Gamma(1)$ that is linear in the index $[\Gamma(1):\Gamma]$, with a constant that only depends on the residue field degree and the degree of the chosen "infinite" place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

**Remarks.** The argument to prove inequality (9.5) in this paper is incorrect; a correct proof is given by Papikian in the Münster J. of Math. **9**.

### Quantum statistical mechanics, L-series and anabelian geometry I: Partition functions

(with Matilde Marcolli) In: Trends in Contemporary Mathematics, INdAM Series, Vol. 8 (2014), pp. 47-57, Springer Verlag.

The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical (QSM) system, built from abelian class field theory. We introduce a general notion of isomorphism of QSM-systems and prove that it preserves (extremal) KMS equilibrium states.
We prove that two number fields with isomorphic quantum statistical mechanical systems are arithmetically equivalent, i.e., have the same zeta function. If one of the fields is normal over $\mathbf Q$, this implies that the fields are isomorphic. Thus, in this case, isomorphism of QSM-systems is the same as isomorphism of number fields, and the *noncommutative* space built from the *abelianized* Galois group can replace the *anabelian* absolute Galois group from the theorem of Neukirch and Uchida.

**Remarks.** This paper is an updated version of part of arxiv:1009.0736 We have split the original preprint into various parts, depending on the methods that are used in them. In the current part, these belong mainly to mathematical physics.

### Curves, dynamical systems, and weighted point counting

Proc. Natl. Acad. Sci. USA **110** (24) (2013), 9669-9673.

Suppose $X$ is a (smooth projective irreducible algebraic) curve over a finite field $k$. Counting the number of points on $X$ over all finite field extensions of $k$ will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over $k$ have the same zeta function (i.e., the same number of points over all extensions of $k$) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet $L$-series of the two curves are equal via an isomorphism of their Dirichlet character groups, then the curves are isomorphic up to “Frobenius twists”, i.e., up to automorphisms of the ground field. Because $L$-series count points on a curve in a “weighted” way, we see that weighted point counting determines a curve. In a sense, the result solves the analogue of the isospectrality problem for curves over finite fields (also know as the “arithmetic equivalence problem”): It states that a curve is determined by “spectral” data, namely, eigenvalues of the Frobenius operator of $k$ acting on the cohomology groups of all $\ell$-adic sheaves corresponding to Dirichlet characters. The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that we make precise.

**Remarks.** The formula at the top of p. 9672 makes no sense and should read $\frac{\Phi}{\Phi(1)} \colon S_X \rightarrow S_Y$.

### Graphs reconstruction and quantum statistical mechanics

(with Matilde Marcolli) J. Geom. Phys. **72** (2013), 110-117.

We study in how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.

### Measure theoretic rigidity for Mumford curves

(with Janne Kool) Ergodic Th. Dyn. Sys. **33** (2013), 851-869.

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve.

**Remarks.** Item (i) of Theorem 1 should start "The intersection dual graphs of the special fibers" instead of "The special fibers".

### Matrix divisbility sequences

(with Jonathan Reynolds)
Acta Arith. **156** (2012), 177-188.

We show that many existing divisibility sequences can be seen as sequences of determinants of matrix divisibility sequences, which arise naturally as Jacobian matrices associated to groups of maps on affine spaces.

### Toroidal automorphic forms, Waldspurger periods and double Dirichlet series

(with Oliver Lorscheid)
in: "Multiple Dirichlet Series L-function and Automorphic Forms",
Progress in Math. **300** (2012), 131-146.

The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a $GL_2$-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight $s$ is toroidal for a given torus precisely if $s$ is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field $F$. We prove that it decomposes into a space spanned by all derivatives up to order $n-1$ of an Eisenstein series of weight $s$ and class group character $\omega$ precisely if $s$ is a zero of order $n$ of the $L$-series $L(\omega,s)$, and a space consisting of exactly those cusp forms the central value of whose $L$-series is zero. The proofs are based on an identity of Hecke for toroidal integrals of Eisenstein series and a result of Waldspurger about toroidal integrals of cusp forms combined with non-vanishing results for twists of $L$-series proven by the method of double Dirichlet series.

### The spectral length of a map between Riemannian manifolds

(with Jan Willem de Jong)
J. Noncommut. Geom. **45** (2012), 721-758.

To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smooth diffeomorphism between Riemannian manifolds is an isometry, in terms of equality along pullback. We also use the invariant to define the (spectral) length of a map between Riemannian manifolds, where a map of length zero between manifolds is an isometry. We show that this length induces a distance between Riemannian manifolds up to isometry.

### Un anneau de déformation universel en conducteur supérieur

(with Jakub Byszewski and Fumiharu Kato)
Proc. Japan Acad. Sci., Ser. A **88** (2012), no. 2, 25-27.

Let $k$ denote a perfect field of characteristic 5. We show that the versal deformation ring of an element of order 5 and Hasse conductor 2 as automorphism of a ring of formal power series $k[[t]]$ computed by Bertin and Mezard, is in fact universal. This provides the first example of a non trivial universal deformation ring in higher conductor.

### Reconstructing global fields using noncommutative geometry

Oberwolfach Report **45** (2011), 27-30.

This is a survey about arxiv:1009.0736.

### Class field theory as dynamical system

in: Arbeitstagung 2011 (Don Zagier 60th birthday conference), MPIM Preprint Series **34** (2011), pp.89-92.

This is the text from a talk at the Arbeitstagung 2011, which can serve as an introduction to arxiv:1009.0736 and arXiv:1007.0907. I first discuss how a global field is determined by a certain dynamical system, and how this relates to abelian L-series determining those fields. I then discuss an analog in Riemannian geometry, and how it leads to a metric in the space of closed Riemannian manifolds.

**Remarks.** For the entire volume, look here. In the function field statement of the preprint version, replace "equivalent to isomorphism of
the curves" by "equivalent to isomorphism of
the curves up to an automorphism of the ground field".

### Relative entropy as a measure of inhomogeneity in general relativity

(with Nikolas Akerblom) J. Math. Phys. **53** (2012) 012502 (10 pp).

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.

### Nonrelativistic Chern-Simons vortices on the torus

(with Nikolas Akerblom, Gerben Stavenga and Jan-Willem van Holten) J. Math. Phys. **52** (2011) 072901 (17pp).

A classification of all periodic self-dual static vortex solutions of the Jackiw-Pi model is given. Physically acceptable solutions of the Liouville equation are related to a class of functions which we term Omega-quasi-elliptic. This class includes, in particular, the elliptic functions and also contains a function previously investigated by Olesen. Some examples of solutions are studied numerically and we point out a peculiar phenomenon of lost vortex charge in the limit where the period lengths tend to infinity, that is, in the planar limit.

### Three examples of the relation between rigid-analytic and algebraic deformation parameters(with Fumiharu Kato and Aristides Kontogeorgis)
Israel J. Math. **180** (2010), 345-370.

**180**(2010), 345-370.

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.

### A compact codimension two braneworld with precisely one brane

(with Nikolas Akerblom) Phys. Rev. D **81** (2010) 124025 (6pp).

Building on earlier work on football shaped extra dimensions, we construct a compact codimension two braneworld with precisely one brane. The two extra dimensions topologically represent a 2-torus which is stabilized by a bulk cosmological constant and magnetic flux. The torus has positive constant curvature almost everywhere, except for a single conical singularity at the location of the brane. In contradistinction to the football shaped case, there is no fine-tuning required for the brane tension. We also present some plausibility arguments why the model should not suffer from serious stability issues.

### Arithmetic equivalence, the Goss zeta function, and a generalisation

(with Aristides Kontogeorgis and Lotte van der Zalm) J. Numb. Th. **130** (2010),no. 4, 1000-1012.

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gassmann-equivalence (a purely group theoretical property), but this statement fails if the degree exceeds the characteristic. We introduce a "Teichmueller lift" of the Goss zeta function and show that equality of such is always the same as Gassmann equivalence.

### Which weakly ramified group actions admit a universal formal deformation?

(with Jakub Byszewski) Ann. Inst. Fourier **59** (2009), no. 3, 877-902.

Consider a formal (mixed-characteristic) deformation functor $D$ of a representation of a finite group $G$ as automorphisms of a power series ring $k[[t]]$ over a perfect field $k$ of positive characteristic. Assume that the action of $G$ is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such $D$ that are not pro-representable occur if $k$ has characteristic two and $G$ is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.

### Toroidal automorphic forms for certain function fields

(with Oliver Lorscheid) J. Numb. Th. **129** (2009), 1456-1463.

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on $GL_2$ is toroidal if all its right translates integrate to zero over all nonsplit tori in $GL_2$, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus $g$ zero or one, and with a rational place. The space has dimension $g$ and is spanned by the expected Eisenstein series. We deduce an "automorphic" proof for the Riemann hypothesis for the zeta function of those curves.

### Zeta functions that hear the shape of a Riemann surface

(with Matilde Marcolli) J. Geom. Phys. **58** (2008), no. 5, pp. 619-632.

To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose "Riemannian" aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)conformal isomorphism class of the corresponding Riemann surface. Thus, you can hear the shape of a Riemann surface, by listening to a suitable spectral triple.

**Remarks.** Section 2 contains a needlessly complicated computation of coefficients of generalized zeta functions. The desired equalities already follow by computing just the constant term as in Theorem 4.5 of this paper.

### Defining the integers in large subrings of number fields using one universal quantifier

(with Alexandra Shlapentokh) J. Math. Sci. **158** (2009), no. 5, 713-726.

Julia Robinson has given a first-order definition of the rational integers $\mathbf Z$ in the rational numbers $\mathbf Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8 variables, and where $F$ is a polynomial. We show that for a large class of number fields, not including $\mathbf Q$, for every $\varepsilon>0$, there exists a set of primes $\cal S$ of natural density exceeding $1−\varepsilon$, such that $\mathbf Z$ can be defined as a subset of the "large: subring $\{x \in K : \mathrm{ord}_{\mathfrak p}x >0, \forall \mathfrak p \not \in \cal S \}$ of $K$ by a formula of the form $(\exists \forall \exists)(F=0)$ where there is only one $\forall$-quantifier, and where $F$ is a polynomial.

**Remarks.** The paper was originally published in: 60th birthday volume for Yuri Matijasevich
"Studies in Constructive Mathematics and Mathematical Logic, Part XI" (ed. Maxim Vserminov),
Записки научных семинаров ПОМИ (Proc. St.-Petersburg Math. Sem.) 358 , pp. 199-223 (2008).

### On the K-theory of graph C*-algebras

(with Matilde Marcolli and Oliver Lorscheid) Acta Appl. Math. **102** (2008), no. 1, pp. 57-69.

We classify graph $C^*$-algebras, namely, Cuntz-Krieger algebras associated to the Bass-Hashimoto edge incidence operator of a finite graph. This is done by a purely graph theoretical calculation of the $K$-theory and the position of the unit therein.

### Noncommutative geometry on trees and buildings

(with Matilde Marcolli, Kamran Reihani and Alina Vdovina) Aspects of Math. E **38**, pp. 73-98, Vieweg Verlag, 2007.

We describe the construction of theta summable and finitely summable spectral triples associated to Mumford curves and some classes of higher dimensional buildings. The finitely summable case is constructed by considering the stabilization of the algebra of the dual graph of the special fiber of the Mumford curve and a variant of the Antonescu-Christensen spectral geometries for AF algebras. The information on the Schottky uniformization is encoded in the spectral geometry through the Patterson-Sullivan measure on the limit set. Some higher rank cases are obtained by adapting the construction for trees.

### Elliptic divisibility sequences and undecidable problems about rational points

(with Karim Zahidi) J. reine und angew. Math. **613** (2007) 1-33

Julia Robinson has given a first-order definition of the rational integers $\mathbf Z$ in the rational numbers $\mathbf Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8 variables, and where $F$ is a polynomial. This implies that the $\Sigma_5$-theory of $\mathbf Q$ is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of $\mathbf Z$ in $\mathbf Q$ with quantifier complexity $\forall \exists$, involving only one universally quantified variable. This improves the complexity of defining $\mathbf Z$ in $\mathbf Q$ in two ways, and implies that the $\Sigma_3$-theory, and even the $\Pi_2$-theory, of $\mathbf Q$ is undecidable (recall that Hilbert's Tenth Problem for $\mathbf Q$ is the question whether the $\Sigma_1$-theory of $\mathbf Q$ is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over $\mathbf Q$ for which one cannot decide whether or not they all have a rational point. The conjecture is related to properties of elliptic divisibility sequences on an elliptic curve and its image under rational 2-descent, namely existence of primitive divisors in suitable residue classes, and we discuss how to prove weaker-in-density versions of the conjecture and present some heuristics.

**Remarks.** You might find this paper refered to by the unspeakable title of its first preprint version "Complexity of undecidable formulae in the rationals and inertial Zsigmondy theorems for elliptic curves". The proof of Theorems 4.2 and 5.3 have to be changed as in Remark 2.3 in this paper.

### Relèvements des revêtements de courbes faiblement ramifiés

(with A. Mézard) Math. Z. **254** (2006) 239-255

Let $X$ be a smooth projective curve over a field of characteristic $p>0$ and $G$ a finite group of automorphism of $X$. Let $n(X,G)$ be the characteristic of the versal equivariant deformation ring $R(X,G)$ of $(X,G)$. When the ramification is weak (i.e., all second ramification groups are trivial),we prove that $n(X,G)$ is $0$ or $p$ and we compute $R(X,G)$.

**Remarks.** Corretions can be found in this BC paper, namely: there is a misprint in the versal deformation of $A_4$ on the bottom of page 251 (Remark 4.2 in BC); in Lemma 3.6 on page 245 of, the argument in the last sentence at the bottom of that page that allows one to conclude an equality of deformation parameters should be replaced by the universality of the versal deformation for Z/p from BC Proposition 2.2 (Remark 4.5 in BC); the proof of Proposition 3.8(ii) on page 250 should be replaced by a direct calculation that, however, leads to the same result(Remark 4.7 in BC).

### Lifting an automorphism to finite characteristic

Rend. Sem. Mat. Univ. Padova **113** (2005) 137-139

Discusses a question about the characteristic of versal deformation rings. Part of the collection "Problems from the workshop on "Automorphisms of Curves" (Leiden, August, 2004)".

### Division-ample sets and the Diophantine problem for rings of integers

(with Thanases Pheidas and Karim Zahidi) J. Théor. Nombres Bordeaux **17** (2005) 727-735

We prove that Hilbert's Tenth Problem for a ring of integers in a number field $K$ has a negative answer if $K$ satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over $K$). We relate division-ample sets to arithmetic of abelian varieties.

### Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven

(with Fumiharu Kato) J. Reine Angew. Math. **589** (2005) 201-236

An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called restrained, if all second ramification groups are trivial (e.g., every group action on an ordinary curve is restrained). When the ramification indices satisfy certain numerical criteria, we construct a degenerating equivariant quasi-projective family to which the given curve belongs, and which in a sense is the unique building block for all such restrained equivariant families that ramify above a fixed set of points. The result is used to inductively study automorphisms of ordinary curves.

### The p-adic icosahedron

(with Fumiharu Kato) Notices Amer. Math. Soc. **52** 720-727 (2005)

This paper establishes an analogy between complex and p-adic orbifolds to develop pictorial representations of "regular polyhedra" in the p-adic world. With beautiful cover art by Bil Casselman.

### Mumford curves with maximal automorphism group

(with Fumiharu Kato) Proc. Amer. Math. Soc. **132** (2004) 1937-1941

The maximal number of automorphisms of a Mumford curve over a non-archimedean valued field of positive characteristic is known. In this note, the unique family of curves that attains this bound in genus not equal to 5,6,7 or 8 is explicitly determined, as is its automorphism group.

**Remark.** One case is missing from the main theorem, see this correction.

### Mumford curves with maximal automorphism group. II. Lamé type groups in genus 5--8

(with Fumiharu Kato) Geom. Dedicata **102** (2003) 127-142

A Mumford curve of genus $g=5,6,7$ or $8$ over a non-archimedean field of characteristic $p$ (such that if $p=0$, the residue field characteristic exceeds $5$) has at most $12(g-1)$ automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.

### Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic

(with Fumiharu Kato) Duke Math. J. **116** (2003) 431-470

We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve $X$ over a field of positive charateristic with prescribed action of a finite group $G$, and the data are computed in terms of the ramification behaviour of $X \rightarrow G\backslash X$. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group $N$ in $\mathrm{PGL}(2,K)$ over a non-archimedean valued field $K$, and the data are computed in terms of the Bass-Serre representation of $N$ via a graph of groups. Finally, if $F$ is a free subgroup of $N$ such that $N$ is contained in the normalizer of $F$ in $\mathrm{PGL}(2,K)$, then the Mumford curve associated to $F$ becomes equipped with an action of $N/F$, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of $N$.

**Remarks.** The arxix preprint version does not contain the correct versal deformations (they are not of order $p$); see the published version for the correct formulas, involving "trunctated formal Chebyshev polynomials".

### The 2-primary class group of certain hyperelliptic curves

J. Number Theory **91** (2001) 174-185

Let $q$ be an odd prime, $e$ a non-square in the finite field $F$ with $q$ elements, $p(T)$ an irreducible polynomial in $F[T]$ and $A$ the affine coordinate ring of the hyperelliptic curve $y^2=ep(T)$ in the $(y,T)$-plane. We use class field theory to study the dependence on $\deg(p)$ of the divisibility by 2, 4, and 8 of the class number ofthe Dedekind ring $A$. Applications to Jacobians and type numbers of certain quaternion algebras are given.

### Two-torsion in the Jacobian of hyperelliptic curves over finite fields

Arch. Math. (Basel) **77** (2001) 241-246; Erratum: Arch. Math. (Basel) **85** loose erratum (2005)

We determine the exact dimension of the $\mathbf{F}_2$-vector space of $\mathbf{F}_q$-rational 2-torsion points in the Jacobian of a hyperelliptic curve over $\mathbf{F}_q$ ($q$ odd) in terms of the degrees of the rational factors of its discriminant, and relate this to genus theory for the corresponding function field. As a corollary, we prove the existence of a point of order $> 2$ in the Jacobian of certain real hyperelliptic curves.

### Discontinuous groups in positive characteristic and automorphisms of Mumford curves

(with Fumiharu Kato and Aristides Kontogeorgis) Math. Ann. **320**(2001) 55-85

A Mumford curve of genus $g$ ($>1$) over a non-archimedean valued field $k$ of positive characteristic has at most $\mathrm{max}\{12(g-1), 2 \sqrt{g} (\sqrt{g}+1)^2\}$ 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 $\mathrm{PGL}(2,k)$ 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. .

**Remarks.** Attention: the main bound in this paper is wrong and the correct theory can be found in Voskuil-van der Put arxiv:1707.03644.

### Diangle groups

in: Proceedings 2000 Kinosaki Symposium on algebraic geometry (2001) 138-143

This is an expository paper on arxiv:math/9908173.

### Nichtarchimedische Geometrie

in: Max-Planck-Gesellschaft: Jahrbuch 2000, Verlag Vandenhoeck & Ruprecht, Goettingen, 566-571.

This is an expository paper on recent research in $p$-adic geometry at the MPIM (the linked file is the English translation).

### Topology of Diophantine sets: remarks on Mazur's conjectures

(with Karim Zahidi) in: Hilbert's Tenth Problem: Relations with arithmetic and algebraic geometry, Contemp. Math. **270** (2000) 253-260.

We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over that finite field. Both proofs depend upon Matijasevich's theorem.

### Zeros of Eisenstein series, quadratic class numbers and supersingularity for rational function fields

Math. Ann. **314** (1999) 175-196

The support of the divisor of certain Eisenstein series on the modular curve for $GL(2,{\bf F}_q[T])$ generate a "big" extension of ${\bf F}_q(T)$. From this, an upper bound for the number of supersingular primes of a fixed degree for a given Drinfeld module can be deduced. In the course of the proof, 2-power divisibility of the class number of certain hyperelliptic extensions of ${\bf F}_q(T)$ is studied.

### Deligne's congruence and supersingular reduction of Drinfeld modules

Arch. Math. (Basel) **72** (1999) 346-353

Most rank two Drinfeld modules are known to have infinitely many supersingular primes. But how many supersingular primes {\sl of a given degree} can a fixed Drinfeld module have? In this paper, a congruence between the Hasse invariant and a certain Eisenstein series is used for obtaining a bound on the number of such supersingular primes. Certain exceptional cases correspond to zeros of certain Eisenstein series with rational $j$-invariants.

### Stockage diophantien et hypothèse abc généralisée

C. R. Acad. Sci. Paris Sér. I Math. **328** (1999) 3-8

Let $(M,L,\phi)$ denote a triple consisting of a first order language $L$ and a set $M$ that admits an interpretation $\phi$ of $L$. We define what it means for a second triple $(M',L',\phi')$ to be positive existential in $(M,L,\phi)$, which generalizes the notion of positive existential set. If the cartesian product $(M\times M,L,\phi\times \phi)$ is positive existential in $(M,L,\phi)$, we say it admits positive existential storing. This generalizes the notion of a storing function known from recursion theory. We then prove that $\mathbf{Z}$ and certain function fields admit positive existential storing.

### Drinfeld modular forms of level T

in: Drinfeld modules, modular schemes and applications World Scientific - Singapore (1997) 272-281.

This paper gives a presentation for the ring of Drinfeld modular forms for the principal congruence group of level $T$ over $\mathbf{F}_q[T]$ by generators and relations.

### A survey of Drinfeld modular forms

in: Drinfeld modules, modular schemes and applications World Scientific - Singapore (1997) 167-187.

This is a survey on the theory of Drinfeld modular forms; We survey the basic definitions, Eisenstein series, Serre-derivatives, Poicaré-series, relations to moduli of Drinfeld modules, the genus of modular curves, modular function fields, dimension formulas for spaces of modular forms, congruences for modular forms and supersingular invariants and the relation to harmonic cochains.

### Drinfeld modular forms of weight one

J. Number Theory **67** (1997) 215-228

This paper studies the ring of Drinfeld modular forms for a principal congruence subgroup of $\mathrm{GL}_2({\bf F}_q[T])$. First, it is shown that Eisenstein series of weight one generate the vector space of modular forms of weight one (which is in sharp contrast with the classical situation, where, e.g., the square of Dedekind's eta-function $\eta^2$ is a cusp form of weight one for $\Gamma(12)$). Using Castelnuovo-Mumford regularity, we show that the ring of modular forms is generated by these Eisenstein series and the cusp forms of weight two, thus improving slightly a bound obtained by D. Goss. We then study embeddings of Drinfeld modular curves via Eisenstein series, and the normality of rings of modular forms. The next paragraph interprets Eisenstein series as torsion points of a generic Drinfeld module, to obtain a reduced set of equations for the image of the embedded modular curve under the group of permutations of the variables. This result will allow the use of computational commutative algebra for solving the original problem. In the final paragraph, we calculate explicitly the degrees of various embeddings of Drinfeld modular curves, and give applications to automorphisms of modular curves and the original problem for linear and quadratic level.

### Sur les zéros des séries d'Eisenstein de poids q^k-1 pour GL(2,F_q[T])

C. R. Acad. Sci. Paris Sér. I Math. **321** (1995) 817-820

If $x$ is a zero of an Eisenstein series of weight $q^k-1$, $k>0$ for $\mathrm{GL}_2(\mathbf{F}_q[T])$ in the Drinfeld upper half plane, we show that its $j$-invariant has degree $q$ or is zero. One can then normalize $x$ such that its degree is $0$. We find that $x$ is elliptic or transcendental over $\mathbf{F}_q(T)$, and all zeros are simple.

# Unpublished

### Torsion of Drinfeld modules and equicharacteristic unimodular Galois covers

with Marina Tripolitaki, preprint (2002)

**Remarks.** The paper has to revised, because there is some sign error in the reciprocity statement.

### Quantum statistical mechanics, L-series and anabelian geometry

(with Matilde Marcolli) preprint (2010), 47pp.

It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a $C^*$-algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch-Uchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is an isomorphism of character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding L-series (not just the zeta function), then the number fields are isomorphic.This is also equivalent to the purely algebraic statement that there exists a topological group isomorphism as a above and a norm-preserving group isomorphism between the ideals of the fields that is compatible with the Artin maps via the other map.

**Remarks.** This paper is split into different parts, each concentrating on one method and one result.

# Other works

### Popularisation/Columns

(with Jan van Neerven) No one had every accused me of proving a theorem before — Interview with Ken Ribet

Nieuw Arch. Wiskd. (5) 18 ??–?? (2017).

Het lustrum van WONDER

Nieuw Arch. Wiskd. (5) 16,
106–107 (2015).

Differentiaalmeetkunde in de getaltheorie

Nieuw Arch. Wiskd. (5) 15, no. 3 188-191 (2014).

DVD Review of Late Style – Yuri I. Manin looking back on a life in mathematics

Notices A.M.S. 60(1) 84-85 (2013).

Book Review: Alexander Masters: Simon, The Genius in my Basement — Het gapend genie

Nieuw Arch. Wiskd. (5) 4 276-277 (2012).

Book Review: A. Terras, Zeta Functions of Graphs: A Stroll through the Garden

Nieuw Arch. Wiskd. (5) 12, nr. 3 (2011)

The flashes of insight never came for free — Interview with Alain Connes.
with Klaas Landsman and Walter van Suijlekom

Nieuw Arch. Wiskd. (5) 11 250-256 (2010)

Book Review Yuri Manin: Mathematics as Metaphor: Ik loop blootsvoets op worstebroodjes

Nieuw Arch. Wiskd. (5) 11 117-118 (2010)

Wiskundig onderzoek per computer? (ed.)

Nieuw Arch. Wiskd. (5) 10 197-201 (2009)

Book Review: Princeton Companion in Mathematics: Waardig gezelschap aan de koffietafel

Nieuw Arch. Wiskd. (5) 10 40-41 (2009)

Het meten van dubbelzinnigheid (Measuring Ambiguity)

Oratie (Inaugural lecture), Universiteit Utrecht (16 januari 2009)

Column: Duitsland-Nederland 1-0: The Architect Sketch, part 2

Nieuw Arch. Wiskd. (5) 9 273 (2008)

Column: België-Nederland 2-0: De laatste der bourbakisten

Nieuw Arch. Wiskd. (5) 9 207 (2008)

Diophantische vergelijkingen: mogelijkheden en onmogelijkheden

Wiskunde-D module 'Wiskunde in Wetenschap' (Betasteunpunt Utrecht) 51 pp. (2008)

Math Girls rule!

Nieuw Arch. Wiskd. (5) 9 108 (2008)

Column: België-Nederland 1-0: Sole meunière

Nieuw Arch. Wiskd. (5) 9 65 (2008)

Diophantische vergelijkingen: Kunnen we het echt?

Nieuw Arch. Wiskd. (5) 7 39-43 (2006)

Meetkunde, meetkundes, en groepen

de Vakidioot (Studievereniging A-Eskwadraat) '05--'06 4-9 (2006)

Diophantische vergelijkingen vanuit de verte bekeken

Nieuwe Wiskrant 4 23-30 (2005)

Book Review: Klaas Landsman: Requiem voor Newton - Turks Fruit der Natuurkunde

Nieuw Arch. Wisk. (5) 6 317-319 (2005)

Over Jean-Pierre Serre - Parti d'un exercice, on se retrouve avec une théorie

Nieuw Arch. Wiskd. (5) 5 35-37 (2004)

Book Review: A. Stubhaug, Ein aufleuchtender Blitz: Niels Henrik Abel und seine Zeit

Nieuw Arch. Wiskd. (5) 5, nr. 2 (2004)

Aritmetische meetkunde

de Vakidioot (Studievereniging A-Eskwadraat) '02--'03 5-7 (2002)

Book Review: D. Haskell, A. Pillay and C. Steinhorn (eds.), Model theory, algebra and geometry

Nieuw Arch. Wiskd. (5) 2, nr. 4 (2001)

Book Review: H. Cartan and S. Eilenberg, Homological algebra (reprint)

Nieuw Arch. Wiskd. (5) 1, nr. 3 (2000)

Book Review: Yu. Manin and S. Gelfand, Methods of homological algebra

Mededelingen Wiskundig Genootschap, review 97.03-18-839 (1997)