Global Oort Groups

TL;DR

This paper characterizes Oort groups for prime p, showing they belong to a specific class of finite groups and establishing conditions under which they lift from characteristic p to characteristic 0.

Contribution

It provides a classification of Oort groups and proves their equivalence to a particular class under a conjecture, with unconditional results for certain group orders.

Findings

All Oort groups are contained in a specific class of finite groups.

Unconditional proof of class equality when group order is not divisible by 2p^2.

Established connections between local and global lifting problems.

Abstract

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all Oort groups lie in a particular class of finite groups that we characterize, with equality of classes under a conjecture about local liftings. We prove this equality unconditionally if the order of G is not divisible by 2p^2. We also treat the local lifting problem and relate it to the global problem.