Deformation data, Belyi maps, and the local lifting problem

TL;DR

This paper investigates the existence of specific differential forms in positive characteristic, utilizing Belyi maps to address the local lifting problem for groups with Sylow p-subgroups of order p.

Contribution

It introduces new existence and nonexistence results for good deformation data and applies reduction modulo p of Belyi maps to solve the local lifting problem.

Findings

Proved existence and nonexistence of certain differential forms in positive characteristic.

Solved the local lifting problem for groups with Sylow p-subgroups of order p.

Utilized reduction modulo p of Belyi maps in the proofs.

Abstract

We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.