Tempered fundamental group and metric graph of a Mumford curve

TL;DR

This paper explores the tempered fundamental group of p-adic algebraic varieties, demonstrating how it encodes the metric structure of Mumford curves and establishing invariance properties and links to Jacobians.

Contribution

It shows that the metric structure of a Mumford curve's stable model graph can be recovered from its tempered fundamental group, introducing new invariance and linking to Jacobians.

Findings

Recovered metric structure from the tempered fundamental group.

Proved birational invariance and invariance under algebraically closed extensions.

Linked the tempered fundamental group of curves to that of their Jacobians.

Abstract

This paper is an attempt to give some general results on the tempered fundamental group of a $p$-adic smooth algebraic varieties (which is a sort of analog of the topologic fundamental group of complex algebraic varieties in the p-adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of this Mumford curve. We will also prove birational invariance, invariance by algebraically closed extensions and a Kuenneth formula for the tempered fundamental group. We will describe the tempered fundamental group of an abelian variety and link the tempered fundamental group of a curve to the tempered fundamental group of its Jacobian variety.