# Number Theory Seminar by Paul Helminck

Paul Helminck (Bremen) will give a Number Theory Seminar on Friday, January 11, in Aud 10 starting 14:15.

Title: Decompositions of tame fundamental groups of nonarchimedean curves using metrized complexes

Abstract:  In this talk, I’ll discuss a natural functor from the category of tame étale coverings of a punctured nonarchimedean curve to the category of tame étale coverings of a metrized complex associated to the punctured curve. In its simplest form, this functor takes a covering of algebraic curves and assigns to it a covering of intersection graphs arising from a morphism of semistable models. By considering the enhanced category of coverings of metrized complexes with gluing data, we then show that we obtain an equivalence of categories, yielding a natural notion of a profinite fundamental group for metrized complexes since the categories involved are Galois categories.

We then use this functor to define the absolute inertia and decomposition groups of subcomplexes of the metrized complex. These inertia and decomposition groups classify the unramified and completely split extensions respectively, just as they did in the number field case. The abelian nonpunctured extensions are then classified by torsion points of the Jacobian and we use the analytic Poincaré-Lelong formula to relate the inertia and decomposition groups to natural subdivisions in the analytic Jacobian. In the discretely valued case, we show that these subdivisions correspond exactly to the ones induced by the identity component and the maximal torus of the Néron model of the Jacobian.