  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.
