Skeletal filtrations of the fundamental group of a non-archimedean curve

TL;DR

This paper develops a categorical framework linking residually tame coverings of non-archimedean curves to tame coverings of associated metrized complexes, enabling new filtrations of the fundamental group and insights into Jacobian extensions.

Contribution

It generalizes semistable reduction for residually tame coverings and establishes an equivalence of categories incorporating gluing data, leading to novel filtrations of the fundamental group.

Findings

Established an equivalence of categories between coverings with gluing data and tame coverings of metrized complexes.

Defined filtrations of the fundamental group, including absolute decomposition and inertia groups.

Proved that certain extensions from the fundamental group coincide with parts of the analytic Jacobian.

Abstract

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first generalize a result by Liu and Lorenzini by proving a simultaneous semistable reduction theorem for residually tame coverings. We then use this to construct a functor from the category of residually tame coverings of a marked curve $(X,D)$ to the category of tame coverings of a metrized complex $\Sigma$ associated to $(X,D)$. We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. Using this equivalence, we then define filtrations of the fundamental group of the marked curve, giving for instance the absolute decomposition and inertia groups of the metrized complex. We then use the analytic slope formula to prove that the extensions that arise from the abelianizations of the decomposition and inertia quotients coincide with the extensions that arise from the toric and connected parts of the analytic Jacobian of the curve.