Cyclic extensions and the local lifting problem

Abstract

The local Oort conjecture states that, if $\Gamma$ is cyclic and $k$ is an algebraically closed field of characteristic $p$, then all $\Gamma$-extensions of $k[[t]]$ should lift to characteristic zero. We prove a critical case of this conjecture. In particular, we show that the conjecture is always true when $v_p(|\Gamma|) \leq 3$ and is true for arbitrarily highly $p$-divisible cyclic groups $\Gamma$ when a certain condition on the higher ramification filtration is satisfied.