The Local Lifting Problem for $D_4$

TL;DR

This paper characterizes D4-extensions in characteristic two, determines their ramification breaks, and proves that all such extensions lift to characteristic zero, establishing D4 as a local Oort group for prime 2.

Contribution

It provides a complete characterization of D4-extensions in characteristic two and proves the local lifting problem has an affirmative solution for these extensions.

Findings

D4-extensions are fully characterized in characteristic two

Ramification breaks of D4-extensions are determined

All D4-extensions lift to characteristic zero

Abstract

For a prime $p$, a cyclic-by-$p$ group $G$ and a $G$-extension $L|K$ of complete discrete valuation fields of characteristic $p$ with algebraically closed residue field, the local lifting problem asks whether the extension $L|K$ lifts to characteristic zero. In this paper, we characterize $D_4$-extensions of fields of characteristic two, determine the ramification breaks of (suitable) $D_4$-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every $D_4$-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that $D_4$ is a local Oort group for the prime 2.