Cyclic Extensions and the Local Lifting Problem

The local Oort conjecture states that, if G is cyclic and k is an algebraically closed field of characteristic p, then all G-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(|G|) \leq 3, and is true for arbitrarily highly p-divisible cyclic groups G when a certain condition on the higher ramification filtration is satisfied.