This paper investigates Galois branched covers of the open p-adic disc and their reductions to characteristic p. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduc...

We consider normal covers of CP with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian squaretiled surfaces, a...

The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified, Treumann, based on unpublished ideas of MacPherson, constructed an ‘exit category’ (in the terminology of this paper, the ‘fundamental category’) which classifies constructible sheaves, equivalently stratified etale covers. This pap...

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u+ and u− in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of u+ at covers of γ agrees with the total multiplicity of the positive ends of u − at covers of γ. However, unlike in the usual g...

We study a family of Teichmüller curves T (n,m) constructed by Bouw and Möller, and previously by Veech and Ward in the cases n = 2, 3. We simplify the proof that T (n,m) is a Teichmüller curve, avoiding the use Möller’s characterization of Teichmüller curves in terms of maximally Higgs bundles. Our key tool is a description of the period mapping of T (n,m) in terms of Schwarz triangle mappings...

We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show unbranched multiple covers closed holomorphic generically regular, and simple index $0$ in dimensions greater than four super-rigid, implying, e.g., Gromov-Witten invariants Calabi-Yau $3$-folds reduce sums local finite sets embedded cu...

In this paper, we discuss the theories of admissible covers (HarrisMumford) and limit linear series (Eisenbud-Harris), two techniques for using degenerations to study maps between curves. While the theories are superficially very similar, they have certain fundamental differences which we will also explore. 1. Motivational problems We begin by laying out a number of questions on curves which sh...

The importance of this result lies primarily in the fact [HLM] that the figure eight knot is universal. That is, that all closed, orientable 3-manifolds are obtainable as branched covers over S, branched over the knot. We also obtain results about incompressible (embedded) tori in some cases. Existence or nonexistence of injectively immersed tori is critical to the understanding of a 3-manifold...