Gunther Cornelissen

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.

Elliptic divisibility sequences and undecidable problems about rational points

Abstract Julia Robinson has given a first-order definition of the rational integers ℤ in the rational numbers ℚ by a formula (∀∃∀∃) (F = 0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ5-theory of ℚ is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of ℤ in ℚ with quantifier complexity ∀∃, involving only one universally quantified variable. This improves the complexity of defining ℤ in ℚ in two ways, and implies that the Σ3-theory, and even the Π2-theory, of ℚ is undecidable (recall that Hilberts Tenth Problem for ℚ is the question whether the Σ1-theory of ℚ is undecidable). In short, granting the conjecture, there is a one-parameter family of hypersurfaces over 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.

Nonrelativistic Chern-Simons vortices on the torus

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 Ω-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.

The spectral length of a map between Riemannian manifolds

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.

Mumford curves with maximal automorphism group

1. IntroductionIt is well-known (cf. Conder [2]) that if a compact Riemann surface of genusg ≥ 2 attains Hurwitz’s bound 84(g − 1) on its number of automorphisms, thenits automorphism group is a so-called Hurwitz group, i.e., a finite quotient of thetriangle group ∆(2,3,7). Equivalently, the Riemann surface is an ´etale cover of theKlein quartic X(7). However, it is hitherto unknown which finite groups can occuras Hurwitz groups. It is even true that, for every integer n, there exists a g such thatthere are more than n non-isomorphic Riemann surfaces of genus g which attainHurwitz’s bound (Cohen, [1]). In this note, we want to show that the correspondingquestions for Mumford curves of genus g over non-archimedean valued fields ofpositive characteristic have a very easy answer: the maximal automorphism groupscan be explicitly described, and they occur for an explicitly given 1-parameterfamily of curves (at least for g /∈ {5,6,7,8}).The set-up for our result is as follows. Let (k,|·|) be a non-archimedean valuedfield of positive characteristic, and X a Mumford curve ([7], [5]) of genus g over k.This means that the stable reduction of X over the residue field ¯k of k is a unionof rational curves intersecting in ¯k-rational points. Equivalently, as a rigid analyticspace over k, the analytification X

Curves, dynamical systems, and weighted point counting

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 ℓ-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.

Relative entropy as a measure of inhomogeneity in general relativity

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.

A Compact Codimension Two Braneworld with Precisely One Brane

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.

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

Abstract. We determine the exact dimension of the