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 vp(|G|) ≤ 3, and is true for arbitrarily highly p-divisible cyclic groups G when a certain condition on the higher ramification filtration is satisfied.

We solve the local lifting problem for the alternating group A 4 , thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A 4 -extension of k[[s]] lifts to characteristic zero.

Let K be a complete discrete valuation field of mixed characteristic (0, p) with algebraically closed residue field, and let f : Y → P 1 be a three-point G-cover defined over K, where G has a cyclic p-Sylow subgroup P . We examine the stable model of f , in particular, the minimal extension K st /K such that the stable model is defined over K st . Our main result is that, if |P | = p n and the center of G has prime-to-p order, then the p-Sylow subgroup of Gal(K st /K) has exponent dividing p n−1 . This extends work of Raynaud in the case that |P | = p.

Colmez conjectured a product formula for periods of abelian varieties with complex multiplication by a field K, analogous to the standard product formula in algebraic number theory. He proved this conjecture up to a rational power of 2 for K/Q abelian. In this paper, we complete the proof of Colmez for K/Q abelian by eliminating this power of 2. Our proof relies on analyzing the Galois action on the De Rham cohomology of Fermat curves in mixed characteristic (0, 2), which in turn relies on understanding the stable reduction of Z/2 n -covers of the projective line, branched at three points.