Formal groups and congruences

Abstract

We give a criterion of integrality of a 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 p -adic analytic formula for the local characteristic polynomial at p p is given. We demonstrate applications of our results to formal group laws attached to L L -functions, Artin–Mazur formal groups of algebraic varieties and hypergeometric formal group laws. This paper was written with the intention to give an explicit and self-contained introduction to the arithmetic of formal group laws, which would be suitable for non-experts. For this reason we consider only one-dimensional laws, though a generalization of our approach to higher dimensions is clearly possible. The ideas of congruences and p p -adic continuity in the context of formal groups were considered by many authors. We sketch the relation of our results to the existing literature in a separate paragraph at the end of the introductory section.