Lifting Elementary Abelian Covers of Curves

TL;DR

This paper investigates the conditions under which elementary abelian Galois covers of algebraic curves in characteristic p can be lifted to characteristic zero, providing combinatorial criteria and analyzing specific cases.

Contribution

It introduces a combinatorial criterion for lifting elementary abelian p-covers based on branch loci and studies coalescence of branch points during lifting.

Findings

Established a criterion for lifting elementary abelian p-covers.

Analyzed branch point coalescence on the special fiber.

Studied lifts of specific (/Z)^3-covers with various conductor types.

Abstract

Given a Galois cover of curves $f$ over a field of characteristic $p$, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is $f$. In this paper, we consider the case where the Galois groups are elementary abelian $p$-groups. We prove a combinatorial criterion for lifting an elementary abelian $p$-cover, dependent on the branch loci of lifts of its $p$-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, we analyze lifts for several families of $(\mathbb{Z}/2)^3$-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.