Explicit p-adic unit-root formulas for hypersurfaces

We prove p-adic continuity of the matrices of coefficients of the logarithm of the Artin-Mazur formal group law associated to a projective hypersurface. The coefficient matrices were computed explicitly by Jan Stienstra in 1987, who also proved they satisfy congruences of Atkin and Swinnerton-Dyer type for a wide class of hypersurfaces, namely double covers of a projective space. In the case when reduction modulo p is a nonsingular hypersurface, Stienstra's congruences imply that eigenvalues of our limiting matrices are eigenvalues of Frobenius of zero p-adic valuation on the middle crystalline cohomology of the reduction of the hypersurface modulo p.