Liftings of Galois Covers of Smooth Curves
Let (C,G) be a smooth irreducible projective curve of genus g over an algebraically closed field k of chararacteristic p>0 and G be a finite group of automorphisms of C. It is well known that here, contrary to the characteristic 0 case, Hurwitz‘s bound |G|[less-than-or-eq, slant] 84(g-1) doesn‘t hold in general; in such cases this gives an obstruction to obtaining a smooth galois lifting of (C,G) to characteristic 0. 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[less-than-or-eq, slant] 2 and (e,p)=1, we shall prove that smooth galois liftings exist over W(k)[$^p^^2$[surd radical]1].
Key Words: Sekiguchi–Suwa theory; order p$^2$ automorphisms of p-adic discs; rigid analytic geometry..