Fundamental groups and good reduction criteria for curves over positive characteristic local fields

TL;DR

This paper introduces an overconvergent rigid fundamental group for varieties over equicharacteristic local fields and applies it to establish a $p$-adic criterion for good reduction of semistable curves, extending classical results.

Contribution

It defines a non-abelian overconvergent fundamental group in positive characteristic and proves an analogue of Oda's theorem relating good reduction to Galois action unramifiedness.

Findings

Defined overconvergent rigid fundamental group for varieties over local fields

Proved $p$-adic analogue of Oda's theorem for good reduction

Connected Galois action unramifiedness with good reduction for curves

Abstract

In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(\varphi,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified.