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In the paper, the structure of the $$ \mathcal{O} $$ K [G]-module F( $$ \mathfrak{m} $$ M ) is described, where M/L, L/K, and K/ℚ p are finite Galois extensions (p is a fixed prime number), G = Gal(M/L), $$ \mathfrak{m} $$ M is a maximal ideal of the ring of integers $$ \mathcal{O} $$ M, and F is a Lubin–Tate formal group law over the ring $$ \mathcal{O} $$ K for a fixed uniformizer π.

Let K be a multidimensional local field with characteristic different from the characteristic of its residue field, c be a unit of K, and Fc(X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module Fc( $$ \mathfrak{M} $$ ) over the maximal ideal of the ring of integers in K. Assume that K contains the group of roots of the isogeny [p m]c(X), which we denote by μ Fc, m . Let be the multiplicative group of Cartier curves and c be the formal analog of the module Fc( $$ \mathfrak{M} $$ ). In the present paper, the formal symbol { ·, · }c : K n ( )× c → μ Fc, m is constructed and its basic properties are checked. This is the first step in the construction of an explicit formula for the Hilbert symbol.

In the present paper, norm series for higher-dimensional Honda formal groups are studied. The norm series satisfying the generalized Steinberg relation with respect to the norm residue symbol are considered for Honda groups over higher-dimensional local fields. Necessary and sufficient conditions for a series to be a norm series are obtained. Bibliography: 6 titles.

In this paper, we study the interaction between transferred Chern classes and Chern classes of transferred bundles. We calculate the algebra \( B{P^{*}}\left( {X_{{h\varSigma p}}^p} \right) \) and show that its multiplicative structure is completely determined by the Frobenius reciprocity. We also give some tables of the initial segments of the formal group law in the Morava K-theory which are often useful in calculations.

In the present paper, norm series for Honda formal groups are studied. The Steinberg relation for the classical Hilbert symbol is generalized to Honda formal groups. Necessary and sufficient conditions for a series to satisfy the generalized Steinberg relation are obtained. In the paper, such series are called norm series. Bibliography: 8 titles.

A generalized norm residue symbol on Lubin–Tate formal groups is studied. The triviality of this symbol in the case where the first argument belongs to the definition field of the formal group is investigated. Explicit formulas for the generalized norm residue symbol ( , ) F,n are used. To this end, a restriction on the expansion of the first argument in powers of a uniformizer is removed. Bibliography: 6 titles.

It is known from the Lubin-Tate theory that a Lubin–Tate formal group can be constructed by its distinguished isogeny, and an arbitrary power series (with a few restrictions) can be taken as such an isogeny. A similar statement is also known for Honda formal groups. In the present article, a similar statement is proved in detail for p-typical formal groups in the so-called small ramification case. It is also proved that as a distinguished homomorphism, in general, one cannot take a polynomial. Bibliography: 6 titles.

The existence of distinguished isogenies is proved for formal groups defined over the ring of integers of a local field with ramification index less than the characteristic of the residue field. This generalizes the result of Demchenko for Honda formal groups. Bibliography: 4 titles.

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles.

Some isomorphism criteria for Lubin–Tate formal groups over the ring of integers of a multidimensional local field are given. Bibliography: 4 titles.

The Steinberg relation for the classical Hilbert symbol is generalized to the Lubin–Tate formal groups. Necessary and sufficient conditions for the series satisfying the generalized Steinberg relation are obtained (in the paper such series are called norm series). Bibliography: 5 titles.

In the present paper, we give an explicit formula for the Hilbert symbol in the case of Honda formal groups. The proof uses the Shafarevich basis and the relationship between the Lubin–Tate formal groups and Honda formal groups. Bibliography: 29 titles.