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arXiv · 2608.01089

Weights in \'etale cohomology over mixed-characteristic local fields and applications to anabelian geometry

Yoshiaki Yamamura
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Abstract

In anabelian geometry, Kummer-faithful fields are expected to be suitable as base fields. In recent years, to characterize Kummer-faithfulness in terms of Galois representations, Ozeki and Taguchi defined a notion of (quasi-)highly Kummer-faithful fields as a variant, and posed the following natural question: Are sub-$p$-adic fields quasi-highly Kummer-faithful? In this paper, by studying the weights of $\ell$-adic \'etale cohomology over mixed-characteristic local fields and discussing the $p$-adic analogue, we give an affirmative answer to this question. Furthermore, we extend the class of base fields from mixed-characteristic local fields to complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, and give equivalent conditions for the vanishing of the coinvariants of $\ell$-adic and of $p$-adic \'etale cohomology. As a result, we show that, for mixed-characteristic complete discrete valuation fields whose residue fields are algebraic extensions of some finite field, Kummer-faithfulness and quasi-high Kummer-faithfulness are equivalent.

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Yoshiaki Yamamura. 2026-08-02. Weights in \'etale cohomology over mixed-characteristic local fields and applications to anabelian geometry. https://arxiv.org/abs/2608.01089

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