Lifting Elementary Abelian Covers of Curves
Jianing Yang

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 over a field of characteristic , the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is . In this paper, we consider the case where the Galois groups are elementary abelian -groups. We prove a combinatorial criterion for lifting an elementary abelian -cover, dependent on the branch loci of lifts of its -cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, we analyze lifts for several families of -covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometric and Algebraic Topology · Polynomial and algebraic computation
