The Local Lifting Problem for D₄

2017/06/12 by Weaver, Bradley
#12F10 #14B12 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 14H37 #Secondary: 13B05

paper · doi:10.48550/arxiv.1706.03751

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 D₄-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D₄-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D₄-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D₄ is a local Oort group for the prime 2.

Related

• The local lifting problem for dihedral groups
• Liftable D₄-Covers
• The refined local lifting problem for cyclic covers of order four
• The local lifting problem for (ℤ/2ℤ)³
• Modified proof of a local analogue of the Grothendieck conjecture