The Arithmetic of Wq(R)

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

The structure of the connected component of the identity of the idele-class group of a number field is given in Artin-Tate [1]. It seems worthwhile to investigate whether this result is the trivial of a theorem in the arithmetic of algebraic tori, i.e. the case corresponding to the multiplicative group Gm (as is true of the unit theorem and the finiteness of the class number). It turns out that the formalism of the theory of tori defined over number fields allows one to generalize Artin's proof line-by-line, obtaining the structure of the identity component of the adele-class group of any such torus. I am grateful to Professor T. Ono for suggesting the problem to me. I give now a summary of the definitions and results that I shall need (for all this, see Ono [3]). I denote by Z, Q, R, and C the rings of respectively integers, rationals, real numbers, and complex numbers, and by T an algebraic torus defined over a number field k. T is the group of rational characters of T. If the dimension of T is d, then T is a free abelian group of rank d [3, ?1.1]. T is split by a finite extension K of k, which I choose to be normal over Q. Then the group TK of points of T rational over K is isomorphic to the product (K*)d of K* with itself d times. Every & 'T is a rational map defined over K; hence if g denotes the galois group of K over k, T has the structure of a g-module. The group Hom(Ti, K*) has the structure of a gmodule, given byfo() = (f(a)), and we have g-isomorphisms

I have developed an algorithm and written a GAP functions pautfreegroup:=function(n) (Finite Presentation of automorphism Groups of Free group of Rank n), and autfreeabeliangroup:=function(n) (Finite Presentation of automorphism Groups of Free abelian group of Rank n). Functions using GAP system for computation of a finite presentation for Aut(F_n ) and GL(n,z) respectively. In order to do that we have given a description for the presentation of Aut(F_n) and GL(n,z), the automorphism groups of free groups and free abelian groups respectively.

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Zr§Zp. We show that in that case, it also has a homomorphism onto a split extension of a free abelian group of rank p — 1 by Zr§Zp, and hence also onto a split extension of a direct sum of p — 1 cyclic groups of order s by the metacyclic group. In many cases, (such as if s is coprime with p), this group can be specified exactly. Otherwise there are a finite number of possibilities. A special case is Perko's result that a homomorphism of a knot group onto S3 = Z2^Z?) lifts to S4 = Z 2ίS)Z3<$(Z2 Θ Z2). As an application we obtain information about the derived series of G. In a final section it is shown how to associate a rational polynomial invariant to every metacyclic representation. 1. Lifting metacyclic representations. Let p be a prime, r a divisor of p — 1 and β a primitive rth root modulo p. Let E — Zj§Zp — (Y, S: γr zn sp = 19Y'1SY= Sβ). Up to isomorphism, the group is independent of β. Let G be the knot group of a knot, AT, in the 3-sphere, S3, and let φ be a homomoφhism of G onto E which takes a meridian, m, of K to YaSh. Then g.c.d. (α, r) — 1, since G is generated by conjugates of m. Setting X — YaSb and eliminating 7, we obtain a presentation

In this paper we describe an algorithm that quickly computes a maximal a \mathfrak {a} -valued lattice in an F F -vector space equipped with a non-degenerate bilinear form, where a \mathfrak {a} is a fractional ideal in a number field F F . We then apply this construction to give an algorithm to compute an a \mathfrak {a} -maximal lattice in a quadratic space over any number field F F where the prime p = 2 p=2 is unramified. We also develop the theory of p \mathfrak {p} -neighbors for a \mathfrak {a} -valued quadratic lattices at an arbitrary prime p \mathfrak {p} of O F \mathcal {O}_F (including when p ∣ 2 \mathfrak {p}\mid 2 ) and prove its close connection to the residual geometry of certain quadrics mod p \mathfrak {p} . Finally we give a well-known application of p \mathfrak {p} -neighboring lattices and exact mass formulas to compute a complete set of representatives for the classes in a given genus of (totally definite) quadratic O F \mathcal {O}_F -lattices.

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

Quasi-pure projective and injective torsion free abelian groups of rank 2

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

Given a number field k with ring of algebraic integers o(k), define an equivalence relation by declaring that two ideals a and b lie in the same class if and only if there exist non-zero elements c and d such that ca = db. These classes form an abelian group called the ideal class group of o(k), abbreviated Cl(o(k)). This group, in some sense, measures how different a ring of algebraic integers is from a principal ideal domain.

Decomposition of integer-valued polynomial algebras

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E*⊗ G induces the Koszul complex and its dual Let H𝒩(m,n,p) be the homology of the top complex at Sym m E*⊗ Sym n G⊗ ∧ p (E*⊗ G) and Hℳ(m,n,p) the homology of the bottom complex at D m E⊗ D n G*⊗ ∧ p (E⊗ G*). It is known that H𝒩(m,n,p)≅ Hℳ(m′, n′, p′), provided m + m′ = g − 1, n + n′ = e − 1, p + p′ = (e − 1)(g − 1), and 1 − e ≤ m − n ≤ g − 1. In this article, we exhibit a complex 𝕐 and explicit quasi-isomorphisms from 𝕐 to two complexes, as described above, for the appropriate choice of parameters, which give rise to this isomorphism. Our quasi-isomorphisms may be formed over the ring of integers; they can be passed to an arbitrary ring or field by base change. All of our work is equivariant under the action of the group GL(E) × GL(G); that is, everything we do is independent of the choice of basis. Knowledge of the homology of the top complex is equivalent to knowledge of the modules in the resolution of the Segre module Segre(e,g, ℓ), for ℓ = m − n. The modules {Segre (e,g, ℓ)|ℓ ∈ ℤ} are a set of representatives of the divisor class group of the determinantal ring defined by the 2 × 2 minors of an e × g matrix of indeterminates. If R is the ring of integers, then the homology H𝒩(m,n,p) is not always a free abelian group. In other words, if R is a field, then the dimension of H𝒩(m,n,p) depends on the characteristic of R. The module H𝒩(m,n,p) is known when R is a field of characteristic zero; however, this module is not yet known over arbitrary fields. The modules in the minimal resolution of the universal ring for finite length modules of projective dimension two are equal to modules of the form H𝒩(m,n,p).