Lifting elementary Abelian covers of curves

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, for $p=2$, we analyze lifts for several families of ${(Z/2)}^3$-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.