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arXiv · 1909.05734

The local theory of unipotent Kummer maps and refined Selmer schemes

L. Alexander Betts·Netan Dogra
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Abstract

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

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L. Alexander Betts, Netan Dogra. 2019-09-12. The local theory of unipotent Kummer maps and refined Selmer schemes. https://arxiv.org/abs/1909.05734

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