The refined local lifting problem for cyclic covers of order four

TL;DR

This paper proves a refined local lifting result for cyclic covers of order four in characteristic two, establishing conditions under which such covers can be lifted to characteristic zero, and explores related phenomena in the moduli space of wildly ramified covers.

Contribution

It demonstrates the existence of lifts for cyclic covers of order four with prescribed sub-cover lifts, supporting Sa{"i}di's refined lifting conjecture in new cases.

Findings

Existence of finite extensions allowing lifts of cyclic covers

Validation of Sa{"i}di's refined lifting conjecture in specific cases

Insights into the structure of moduli space of wildly ramified Galois covers

Abstract

Suppose $\phi$ 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 $\phi$'s $\mathbb{Z}/2$-sub-cover to a complete discrete valuation ring $R$ in characteristic zero. We show that there exist a finite extension $R'$ of $R$, which is determined by $\Phi_1$, and a lift $\Phi$ of $\phi$ to $R'$ whose $\mathbb{Z}/2$-sub-cover isomorphic to $\Phi_1 \otimes_R R'$. That result gives a non-trivial family of cyclic covers where Sa{\"i}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.