Local Oort groups and the isolated differential data criterion

TL;DR

This paper improves the computational methods for lifting branched G-covers of curves from characteristic p to zero, confirming the conjecture for specific dihedral groups by analyzing differential forms.

Contribution

It introduces a more efficient computational procedure for meromorphic differential forms, enabling verification of the lifting conjecture for D_25 and D_27 covers.

Findings

All D_25-covers lift to characteristic zero.

All D_27-covers lift to characteristic zero.

Enhanced computational approach for differential forms.

Abstract

It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.