The local theory of unipotent Kummer maps and refined Selmer schemes

We study the Galois action on paths in the ${\mathbb{Q}}_{\mathrm{\ell }}$-pro-unipotent 'etale fundamental groupoid of a hyperbolic curve $X$ over a $p$-adic field with $\mathrm{\ell }\mathrm{\ne }p$. 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.