Higher Hasse--Witt matrices

TL;DR

This paper establishes p-adic congruences for coefficients of polynomial powers, linking them to Gauss--Manin connections, Frobenius operators, and formal group laws, advancing understanding of algebraic structures in characteristic zero.

Contribution

It introduces new p-adic congruences for polynomial coefficients and connects them to geometric and algebraic structures like formal group laws and unit-root crystals.

Findings

Derived p-adic limit formulas for polynomial coefficients.

Linked congruences to Gauss--Manin connection and Frobenius operator.

Associated formal group laws with polynomial functions under certain conditions.

Abstract

We prove a number of p-adic congruences for the coefficients of powers of a multivariate polynomial f(x) with coefficients in a ring R of characteristic zero. If the Hasse--Witt operation is invertible, our congruences yield p-adic limit formulas which conjecturally describe the Gauss--Manin connection and the Frobenius operator on the unit-root crystal attached to f(x). As a second application, we associate with f(x) formal group laws over R. Under certain assumptions these formal group laws are coordinalizations of the Artin--Mazur functors. (This is a final version which we send for a publication.)