Links between associated additive Galois modules and computation of H for local formal group modules

Using the methods described in the papers (Documenta Math. 5 (2000) 657; Local Leopoldt's problem for ideals in p-extensions of complete discrete valuation fields, to appear), we prove that a cocycle for a formal group in a Galois p-extension of a complete discrete valuation field is a coboundary if and only if the corresponding group algebra elements increase valuations by a number that is sufficiently large. We also calculate the valuation of the splitting element of a coboundary. A special case of the main theorem allows us to determine when a p-extension of a complete discrete valuation fields contains a root of a Kummer equation for a formal group. The theorem of Coates-Greenberg for formal group modules in deeply ramified extensions is generalized to noncommutative formal groups. Some results concerning finite torsion modules for formal groups are obtained.     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

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.     

On the Buchstaber formal group law and some related genera

We calculate some formal group laws and genera closely related to the universal Buchstaber formal group law \(\mathcal{F}_B\) .     

Arithmetic of a group of points of a formal group

For a formal group of finite height over a non-ramified extension of a ring of p-adic integers we construct a system of generators of a formal module that is the generalization of the canonic Shafarevich basis and the system of Henniart generators for Lubin — Tate groups.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 9–23, 1991.     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Realization of Abel's universal formal group law

To a complex oriented cohomology theory one may assign a formal group law over . The purpose of this paper is to show that the converse holds true for the case of Abel's universal formal group law , i.e. we will prove the existence of a complex oriented cohomology theory whose formal group law is isomorphic to . Received: 16 July 1998; in final form: 6 November 2000 / Published online: 19 October 2001     

Buchstaber formal group and elliptic functions of small levels

Mathematical Notes - In the paper, we suggest a method for finding relations concerning series defining the Buchstaber formal group. This method is applied to the cases in which the exponent of the...     

Finitely additive set functions into a topological group

The properties of finitely additive set functions defined on a ring S with values in an Abelian topological group are studied. In particular, a condition under which the property of uniform absence of the escaping load (UAEL) is preserved under the extension of a set function from a ring R to an extending ring S R is established.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 847–854, June, 1977.     

On Drinfeld's universal formal group over thep-adic upper half plane

Conclusion All of our results are stated for 2-dimensional modules with action by the quaternion division algebra overQ _p . Drinfeld's results are true in much greater generality. We remark that our results generalize easily to the case of 2-dimensional modules with action by quaternion algebras over extensions ofQ _p by applying the theory of formal -modules. We suspect that Drinfeld's higher dimensional modules over are determined by formulae similar to that in Theorem 46, but with and generalized to moduli for higher dimensionalQ _p -subspaces of ; however, we have not investigated this in any detail.Although this work amplifies Drinfeld's original paper by supplying many details in certain cases, it is seriously limited in that it considers lifts of SFD modules to unramified rings only. The most interesting points in thep-adic upper half plane are the points defined over ramified rings, which reduce modp to the singular points on the special fiber. What happens there? We do not have a simple answer.Drinfeld's moduli for formal groups on thep-adic upper half plane is the basis for his proof that Shimura curves havep-adic uniformizations. In a later work, we hope to exploit improved versions of the techniques in this work to better understand the arithmetic of Shimura curves. In particular, in the course of work onp-adicL-functions, we have been led to construct certain p-adic periods associated to the cohomology of sheaves on Shimura curves which depend essentially on the existence of ap-adic uniformization. We hope to use Drinfeld's moduli to obtain a more natural construction of these periods in terms of the Gauss-Manin connection, and thereby to gain a better understanding of how they might come to appear in special values ofp-adicL-functions.This research was partially supported by an NSF Postdoctoral Fellowship