The local lifting problem for A_4

TL;DR

This paper proves that the alternating group A_4 satisfies the local lifting problem in characteristic 2, showing all A_4-extensions of formal power series rings lift to characteristic zero, impacting the lifting of covers of curves.

Contribution

It demonstrates that A_4 is a local Oort group by proving all its extensions lift from characteristic 2 to zero, solving a longstanding problem.

Findings

All A_4-extensions of k[[s]] lift to characteristic zero.

Every A_4-branched cover of smooth projective curves lifts to characteristic zero.

A_4 is confirmed as a local Oort group.

Abstract

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. As a consequence, every A_4-branched cover of smooth projective curves in characteristic 2 lifts to characteristic zero.