Characterising local fields of positive characteristic by Galois theory and the Brauer group

TL;DR

The paper addresses identifying local fields of characteristic $p$ from Galois data augmented by $p$-torsion in the Brauer group. It combines the absolute Galois group with a bilinear pairing into $\mathrm{Br}(K)[p]$, recasting cohomology via Milnor $K$-theory and Kato cohomology to encode arithmetic information. The main result shows that fields $K$ with $\mathrm{G}_K \cong \mathrm{G}_{\mathbb{F}_q((t))}$, imperfection exponent at most $1$, $\mathrm{Br}(K)[p] \cong \mathbb{Z}/p$, and a canonical pairing are isomorphic to $\mathbb{F}_q((t))$, with a strengthened version using Galois quotients. This work strengthens anabelian-type reconstructions in positive characteristic by illustrating how Brauer-group data and Galois-quotient information together fix the discrete valuation and completeness, yielding a precise local-field identification.

Abstract

We show that each local field $\mathbb{F}_q((t))$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most $1$ by (certain small quotients of) its absolute Galois group together with natural axioms concerning the $p$-torsion of its Brauer group. This complements previous work by Efrat-Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic $p$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Remark 3.1
• Proposition 3.2