S. V. Vostokov

Local fields and their extensions

Complete discrete valuation fields Extensions of discrete valuation fields The norm map Local class field theory I Local class field theory II The group of units of local number fields Explicit formulas for the Hilbert symbol Explicit formulas for the Hilbert pairing on formal groups The Milnor

Ideals of an abelian p-extension of a local field as Galois modules

K

An Explicit Formula of the Hilbert Pairing for Honda Formal Groups

-groups of a local field Bibliography List of notations Index.

Norm Series for Lubin–Tate Formal Groups

Ideals of the ring of integers of an Abelian p-extension of a local field (a finite extension of the field of p-adic numbers) are studied as modules with operators from the Galois group of the extension. Necessary and sufficient conditions are found for the decomposability of these ideals as Galois modules. When the decomposability conditions are satisfied, the decomposition of the ideal into indecomposable summands is found.

Extensions with Almost Maximal Depth of Ramification

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.

Explicit constructions and the arithmetic of local number fields

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.

Hilbert pairing on Lorentz formal groups

The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|DL/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, DL/K is the different of L/K. This condition means that the depth of ramification in L/K has its “almost maximal” value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.

Degeneration of the Hilbert pairing in formal groups over local fields

This is a survey of results obtained by members of the St. Petersburg school of local number theory headed by S.V. Vostokov during the past decades. All these results hardly fit into the title of the paper, since they involve a large circle of ideas, which are applied to an even larger class of problems of modern number theory. The authors tried to cover at least a small part of them, namely, those related to the modern approach to explicit expressions of the Hilbert symbol for nonclassical formal modules in the one- and higher-dimensional cases and their applications in local arithmetic geometry and ramification theory.

Lutz filtration as a Galois module

In this paper, we construct an explicit pairing in Cartier series for formal Lorentz groups of the form (X + Y + XY)/(1 + c2XY), where c is a unit of the ring of integers of the local field. We prove the basic properties of this pairing, namely, bilinearity and invariance, which make it possible to explicitly construct the generalized Hilbert symbol for formal Lorentz groups over rings of integers of local fields with the use of the obtained pairing.

Pairings in local fields and cryptography

For an arbitrary local field K (a finite extension of the field Qp) and an arbitrary formal group law F over K, we consider an analog cF of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing cF has a certain fundamental symbol property for all Lubin–Tate formal groups, then cF = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring OK of integers of K and have a fixed ring O0 of endomorphisms, where O0 is a subring of OK. We prove that if the symbol cF has the above-mentioned symbol property, then cF = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.