Compositio Mathematica

Research Article

Oort groups and lifting problems

T. Chinburga1, R. Guralnicka2 and D. Harbatera3

a1 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA (email: ted@math.upenn.edu)

a2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA (email: guralnic@usc.edu)

a3 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA (email: harbater@math.upenn.edu)

Abstract

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

(Received September 18 2007)

(Accepted January 09 2008)

2000 Mathematics subject classification

  • primary 12F10;
  • 14H37;
  • 20B25;
  • secondary 13B05;
  • 14D15;
  • 14H30

Keywords

  • curves;
  • automorphisms;
  • Galois groups;
  • characteristic p;
  • lifting;
  • Oort conjecture