The Local Lifting Problem for $D_4$

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.

Minor corrections to Sections 5.2 and 6.1. 26 pages. Comments welcome

References in corpus (1)

• Lifting of curves with automorphisms

Cited by in corpus (1)

• Lifting of curves with automorphisms