Formal groups and congruences

TL;DR

This paper establishes a criterion for the integrality of one-dimensional formal group laws using coefficient congruences, and applies it to various mathematical contexts including L-functions and hypergeometric laws.

Contribution

It introduces a new integrality criterion based on coefficient congruences and provides p-adic formulas for local characteristic polynomials, with applications to several areas.

Findings

Criterion for integrality of formal group laws

p-adic formula for local characteristic polynomial

Applications to L-functions, Artin--Mazur groups, hypergeometric laws

Abstract

We give a criterion of integrality of an one-dimensional formal group law in terms of congruences satisfied by the coefficients of the canonical invariant differential. For an integral formal group law a p-adic analytic formula for the local characteristic polynomial at p is given. We demonstrate applications of our results to formal group laws attached to L-functions, Artin--Mazur formal groups of algebraic varieties and hypergeometric formal group laws. (This is the final version submitted to a journal; new parts are added in the section on hypergeometric formal group laws.)