Rational endomorphisms of Fano hypersurfaces

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 \(d\ge \lceil 5(n+3)/6\rceil \) are not birational to Jacobian fibrations of dimension one. A key part of the argument is to resolve singularities of general \(\mu _{p}\)-covers in mixed characteristic p.