Weights in étale cohomology over mixed-characteristic local fields and applications to anabelian geometry

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 étale 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 étale 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.

24 pages