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

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 $(\phi ,\nabla )$-module over the bounded Robba ring ${\mathcal{E}}_K^{†}$, 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.

arXiv comment34 pages, comments very welcome!