Explicit p-adic unit-root formulas for hypersurfaces

TL;DR

This paper establishes p-adic continuity properties of matrices related to the formal group law of hypersurfaces, revealing congruences that connect eigenvalues to Frobenius actions in crystalline cohomology.

Contribution

It provides explicit p-adic formulas for unit roots of hypersurfaces, extending previous congruence results and linking formal group laws to Frobenius eigenvalues.

Findings

Matrices exhibit p-adic continuity in their coefficients.

Entries satisfy Atkin and Swinnerton-Dyer type congruences.

Eigenvalues correspond to Frobenius eigenvalues with zero p-adic valuation.

Abstract

We prove a statement on p-adic continuity of matrices of coefficients of the logarithm of the Artin-Mazur formal group law associated to the middle cohomology of a hypersurface. As Jan Stienstra discovered in 1986, the entries of these matrices are coefficients of powers of the equation of the hypersurface, and in certain cases they satisfy congruences of Atkin and Swinnerton-Dyer type. These congruences imply that eigenvalues of our limiting matrices are eigenvalues of Frobenius of zero p-adic valuation on the middle crystalline cohomology of the fibre at p.