Hazewinkel functional lemma and formal groups classification

Abstract

There are two approaches to the construction of formal groups. The functional lemma proved by Hazewinkel allows to make formal groups with coefficients from a ring of zero characteristic by means of the functional equations using a certain ideal of this ring, overfield and a ring homomorphism with certain properties (for example, identical, and for a local field the Frobenius homomorphism can be chosen). There is a convenient criterion for the isomorphism of formal groups constructed by Hazewinkel’s formula, as well as a formula for logarithms (in particular, the Artin — Hasse logarithm). At the same time, Lubin and Tate construct formal groups over local fields using isogeny, and Honda construct formal groups over discrete normalized ring of characteristic fields, introduces a certain noncommutative
ring induced by the original ring and a fixed homomorphism. The paper establishes a connection between the classical classification of formal groups (standard, generalized and relative formal Lubin — Tate groups and formal Honda groups) and their classification using the Hazewinkel functional lemma. For each type, the corresponding functional equations are compiled and logarithms are studied, as well as series that use for construction the Hilbert symbol explicit formula.

Keywords:

local fields, formal groups, Hilbert symbol, Hazewinkel classification