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|[les ] 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[les ] 2 and (e,p)=1, we shall prove that smooth galois liftings exist over W(k)[$^p^^2$√1].

Let G be a complex linear algebraic group and ρ:G → GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character ϕ ∈ Hom (G, C$^×$), meaning that f ˆ ρ (g) =ϕ (g) f for all g in G. Choose a non-zero relative invariant f$^v$ in C[V$^v$], with character ϕ$^-1$, for the dual representation ρ$^v$:G → GL(V$^v$). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sato says that the Fourier transform of |f|$^s$equals |f$^v$|$^-s$ up to some factors. The purpose of the present paper is to study a finite field analogue of Sato‘s theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field F$_q$ is large enough. Now |f|$^s$ is replaced by χ (f), with χ in Hom (F$_q$$^×$, C$^×$), and the factors involve Gauss sums, the Bernstein–Sato polynomial b(s) of f, and the parity of the split rank of the isotropy group at v$^v$∈ V$^v(F<math>_q$). We also express this parity in terms of the quadratic residue of the discriminant of the Hessian of log f$^v$ (v$^v$). Moreover we prove a conjecture of N. Kawanaka on the number of integer roots of b(s).