Explicit p-adic unit-root formulas for hypersurfaces

2015/01/18 by Masha Vlasenko, Vlasenko, Masha · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1501.04280

arxiv created 2015/01/18 · openalex publication_date 2015/01/18 · arxiv updated 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Higher Hasse--Witt matrices

Related

• A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix
• Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values
• Bounding slopes of p-adic modular forms
• Cohomology and congruences
• p-adic boundary values