Integer valued polynomials and Lubin-Tate formal groups

If R is an integral domain and K is its field of fractions, we let Int(R) stand for the subring of K[x] which maps R into itself. We show that if R is the ring of integers of a p-adic field, then Int(R) is generated, as an R-algebra, by the coefficients of the endomorphisms of any Lubin-Tate group attached to R.     

Hilbert symbol for Lubin-Tate formal groups. II

Explicit formulas are obtained for the generalized Hilbert symbol on the group of points of a formal Lubin-Tate group, simultaneously covering the cases of even and odd characteristics of the residue field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 85–96, 1983.     

Relative Lubin-Tate formal groups and Galois module structure

Some Galois modules attached to divisions fields of relative Lubin-Tate formal groups are studied. Their associated order and module generators are determined.     

The Hilbert symbol for Lubin-Tate formal groups. I

For the Hilbert Symbol on the group of points of a Lubin-Tate formal group there is found a modification of the explicit formulas that covers both the cases of an even and an odd prime p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 77–95, 1982.     

An explicit form of the Hilbert pairing for the relative formal Lubin-Tate groups

In the present paper an explicit formula is derived for the Hilbert symbol on the relative formal Lubin-Tate groups. This formula is a generalization of the Vostokov formula for the ordinary Lubin-Take groups. Bibliography:3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 41–44.     

On singular formal deformations

We discuss the notion of singular formal deformation in algebraic setup. Such deformations show up in both finite and infinite dimensional structures. It turns out that there is a stronger version of singular deformation—called essentially singular—which arises from a singular curve in the base of the versal deformation.     

Canonical decomposition in the group of points of a formal Lubin-Tate group

In the group of points of a formal Lubin-Tate group over a local field there is constructed a canonical decomposition needed for giving an explicit formula for the Hilbert symbol. Here all arguments are independent of the parity of the characteristic of the residue field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 103, pp. 52–57, 1980.     

The Lubin-Tate theory for formal power series fields with finite coefficient fields

The Lubin-Tate theory for formal complex multiplication can be proved by the results in local class field theory on abelian p-extensions in characteristic p, in the case where the ground fields are formal power series fields with finite coefficient fields.     

Krichever formal groups

On the basis of the general Weierstrass model of the cubic curve with parameters μ = (μ₁, μ₂, μ₃, μ₄, μ₆), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied. As a consequence, results on the elliptic Hirzebruch genus with values in ℤ[μ] are obtained.     

Commutative formal groups

The formulations of several results on the structure of commutative formal groups of any dimensional over rings without -torsion are given, and also a geometric construction of a class of formal groups arising from algebraic manifolds is given. Proofs are only outlined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 117–124, 1976.     

Fragmented deformations of primitive multiple curves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y _red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.     

Induced formal deformations and the Cohen-Macaulay property

The main result states: if is a module finite extension of excellent local normal domains which is unramified in codimension two and if represents a deformation of the completion of , then there is a corresponding -algebra deformation such that the ring homomorphism represents a deformation of . The main application is to the ascent of the arithmetic Cohen-Macaulay property for an étale map of smooth projective varieties over an algebraically closed field.     

Compact deformations of Fuchsian groups

A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian groupG if Φ has a quasiconformal extension to the planeh which conjugatesG to a Kleinian group G′ and the dilatation ofh is compactly supported moduloG. We show that for such deformations δ = dim(∧(G′)) = dim(∧_c(G′)) (if δ ≥1) and the image of ∧_e = ∧ ∧_c is contained in a countable union of rectifiable curves and has zero length iffG is divergence type. The first author is partially supported by NSF Grant DMS 01-03626. The second author is partially supported by NSF grant DMS-99-71311.