Canonical Valuations and the Birational Section Conjecture

TL;DR

This paper introduces a generalized notion of canonical valuations for fields, enabling the recovery of $p$-adic valuations from Galois group quotients, and applies this to prove versions of the birational section conjecture over $p$-adic fields.

Contribution

It generalizes the concept of canonical valuations and demonstrates their use in recovering valuations from Galois groups, leading to progress on the birational section conjecture.

Findings

$p$-adic valuations can be recovered from Galois group quotients.

Canonical $ ext{C}$-henselian valuations generalize existing valuation concepts.

Several versions of the birational section conjecture are proved for $p$-adic varieties.

Abstract

We develop a notion of a `canonical $\mathcal{C}$-henselian valuation' for a class $\mathcal{C}$ of field extensions, generalizing the construction of the canonical henselian valuation of a field. We use this to show that the $p$-adic valuation on a finite extension $F$ of $\mathbb{Q}_p$ can be recovered entirely (or up to some indeterminacy of the residue field) from various small quotients of $G_F$, the absolute Galois group of $F$. In particular, it can be recovered fully from the maximal solvable quotient. We use this to prove several versions of the birational section conjecture for varieties over $p$-adic fields.