Crew’s Euler Characteristic Formula Fails for Nonzero Slopes

with respect to étale cohomology with compact supports. This relation fails in general for l = p, but holds in an important special case. Suppose f is a Galois cover and deg f is a power of p. Then the relation χc(X,Qp) = (deg f)χ(Y,Ql) holds; for complete curves, this is due to Shafarevich [2], for arbitrary curves it is equivalent to the Deuring-Shafarevich formula, and in general it is due to in dimension 1 and by Crew [1] in general. For X and Y smooth and proper over Spec k, this assertion can be reinterpreted in terms of crystalline cohomology. Namely, in this case, the crystalline cohomology groups H i crys(X/W ) are finitely generated modules over the Witt ring W of k with a semilinear endomorphism F (the Frobenius). For any rational number λ, we let hλ(X) be the multiplicity of the slope λ in H i crys(X/W ); then h i 0(X) = dimH i c(X,Ql). If we put