The local theory of unipotent Kummer maps and refined Selmer schemes

Abstract

We study the Galois action on paths in the ${\mathbb{Q}}_{\ell }$-pro-unipotent étale fundamental groupoid of a hyperbolic curve $X$ over a $p$-adic field with $\ell \ne p$. We prove an Oda--Tamagawa-type criterion for the existence of a Galois-invariant path in terms of the reduction of $X$, as well as an anabelian reconstruction result determining the stable reduction type of $X$ in terms of its fundamental groupoid. We give an explicit combinatorial description of the non-abelian Kummer map of $X$ in arbitrary depth, and deduce consequences for the non-abelian Chabauty method for affine hyperbolic curves and for explicit quadratic Chabauty.

123 pages