Equidistant liftings of elementary abelian Galois covers of curves

Abstract

In this paper, we discuss the local lifting problem for the action of elementary abelian groups. Studying logarithmic differential forms linked to deformations of $({\mu }_p{)}^n$-torsors, we show necessary conditions on the set of ramification points in order to get equidistant liftings. Such conditions of combinatoric nature lead us to show new obstructions to lifting actions of Z/3Z x Z/3Z.

A crucial mistake was found in Proposition 3.8 that invalidated the main result of this paper. Many thanks to Michel Matignon who pointed this out to me. A corrected discussion of spaces $L_{15,2}$ has now appeared in Section 6.2 of arXiv:2503.19533 as part of a much wider discussion of spaces $L_{m+1,n}$