A p-adic nonabelian criterion for good reduction of curves

Abstract

Let $K$ be a complete discrete valuation field of characteristic $0$, with valuation ring ${\mathcal{O}}_K$ and perfect residue field $k$ of positive characteristic $p$. We prove that a proper and smooth curve over $K$, admitting a semistable model over ${\mathcal{O}}_K$, has good reduction if and only if its unipotent $p$-adic étale fundamental group is crystalline.