  We show that the degrees of rational endomorphisms of very general complex
Fano and Calabi-Yau hypersurfaces satisfy certain congruence conditions by
specializing to characteristic p. As a corollary we show that very general
n-dimensional hypersurfaces of degree $d\ge 5\lceil (n+3)/6\rceil$ are not
birational to elliptic fibrations. A key part of the argument is to resolve
singularities of general p-cyclic covers in mixed characteristic p.
