Liftings of Galois Covers of Smooth Curves, by Barry Green and Michel Matignon

This paper has appeared in Compositio Mathematica, 113, no. 3 (July 1998), 239-274, and so the dvi version has been removed. Let (C,G) be a smooth integral proper curve of genus g over an algebraically closed field k of chararacteristic p>0 and G be a finite group of automorphisms of C. In this paper we study the problem of obtaining a smooth galois lifting of (C,G) to characteristic 0. It is well known that over a field of characteristic p, contrary to the characteristic 0 case, Hurwitz's bound |G| \leq 84(g-1) doesn't hold in general; in such cases this gives an obstruction to obtaining a smooth galois lifting.

We shall give new obstructions of local nature to the lifting problem, even in the case where G is abelian. In the case where the inertia groups are p^ae-cyclic with a \leq 2 and (e,p)=1, we shall prove that smooth galois liftings exist over W(k)[\root p^2 \of 1].



Barry Green <bwg@land.sun.ac.za>
Michel Matignon <matignon@math.u-bordeaux.fr>