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

Cyclic Extensions and the Local Lifting Problem