Cyclic Extensions and the Local Lifting Problem

TL;DR

This paper proves a significant case of the local Oort conjecture, demonstrating that cyclic G-extensions of formal power series rings over algebraically closed fields lift to characteristic zero under specific conditions.

Contribution

It establishes the validity of the local Oort conjecture for cyclic groups with p-divisibility up to 3 and under certain ramification conditions for higher divisibility groups.

Findings

Proves the conjecture for v_p(|G|) ≤ 3.

Validates the conjecture for highly p-divisible cyclic groups with ramification conditions.

Extends understanding of lifting problems for cyclic extensions.

Abstract

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.