The refined local lifting problem for cyclic covers of order four

Abstract

Suppose $\varphi$ is a $\mathbb{Z}/4$-cover of a curve over an algebraically closed field $k$ of characteristic $2$, and ${\Phi }_1$ is a \emph{nice} lift of $\varphi$'s $\mathbb{Z}/2$-sub-cover to a complete discrete valuation ring $R$ in characteristic zero. We show that there exist a finite extension $R^{\prime }$ of $R$, which is determined by ${\Phi }_1$, and a lift $\Phi$ of $\varphi$ to $R^{\prime }$ whose $\mathbb{Z}/2$-sub-cover isomorphic to ${\Phi }_1{\otimes }_R{R}^{\prime }$. That result gives a non-trivial family of cyclic covers where Sa{ï}di's refined lifting conjecture holds. In addition, the manuscript exhibits some phenomena that may shed some light on the mysterious moduli space of wildly ramified Galois covers.

arXiv commentThere were some gaps in section 4 of the previous version. The current result is now weaker than a known one but uses a different approach. We may consider replacing this manuscript with another one that yields a stronger result

References in corpus (1)

• Hurwitz trees and deformations of Artin-Schreier covers