Realizing profinite reduced special groups

Some model-theoretic results in the algebraic theory of quadratic forms

In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of "equivalence"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K 0 , K 1 and K 2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshall's conjecture on signatures, which relies heavily on this theory.

Special groups Pfister forms, saturated subgroups and quotients The space of orders of a reduced group. Duality Boolean algebras and reduced special groups Embeddings Special groups of continuous functions Horn-Tarski and Stiefel-Whitney invariants Algebraic K-theory of fields and special groups Marshall's conjecture for pythagorean fields The category of special groups Some model theory of special groups Appendix A. The universal theory of reduced special groups Appendix B. Table of references for [DM1] and [DM2] Bibliography Index.

It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. They are the subject of this chapter. The investigation of Pfister forms is based essentially on the use of transcendental extensions of the ground field: One considers quadratic forms Σ a ij X i X j . with indeterminates X i . This approach leads naturally to the important concept of the function field of a quadratic form which is the function field of the quadratic Σ a ij X i X j =0.

Powers of the Fundamental Ideal in the Witt Ring

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

Tais work is a development of some logical-categorias aspects of the theory of Special Groups a first order presentation of the algebraic theory of quadratic forma. We study the Profinite Special Groups certan kinds of limita that the Special Groups category has. We build a functor from the category RSG of reduced special groups with SG-morphisms to the category RSG./ of profirJite reduced special groups with continuous SG'-morphisms. We veiify that this functor deserves the title of "profinite hull functor " : it is the left adjoint of the inclusion RSG..r --> RSG. We analyze the behavior of tais furlctor by categorias constructlons : we show that it preserves quotients and complete embeddings . We present the concept of a subformreflecting morphism between special groups arld we show that the natura! transformation that is the unir of suco adjunctior] is composed by this kir)d of morphisms. We identify tais airow in the case of the special group that Game from a boojean algebra with the boolean algebras morphism that generates the topology on the respective Stone space and explore suco identification.

We provide a new proof of the Arason–Pfister Hauptsatz (APH) in the setting of reduced special groups, as developed by Dickmann and Miraglia. Our approach avoids the use of Boolean invariants and instead relies on a construction inspired by Marshall’s quotient, suitably adapted to the context of special groups. We establish structural properties of this quotient and show that it generalizes the Pfister quotient by a Pfister subgroup. Using this framework, we define iterated quadratic extensions of special groups and develop a theory of Arason–Pfister sequences. These tools allow us to prove that any anisotropic form φ∈In(G) over a reduced special group G satisfies the inequality dim(φ)≥2n, where In(G) denotes the n-th power of the fundamental ideal of the Witt ring of G. Our methods are purely algebraic and internal to the theory of special groups, contributing with novel tools to the categorical study of abstract theories of quadratic forms.

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

It is known that the twist space of a (plunging) constant-velocity (CV) coupling with intersecting shafts consists, in all configurations, of a planar field of zero-pitch screws, namely Hunt's fourth special three system (with principal pitches (1; 0; 0)), hereafter denoted S_(3,4). The input and output shafts form a mirror symmetry about the system plane, hereafter referred to as the bisecting plane. The persistence of the twist space, i.e. the invariance of its principal and reciprocal principal pitches, is achieved by a parallel mechanism with prescribed connecting chains, with each one of them having mirror symmetric joint axes about the bisecting plane. Recently, the first author reported an important discovery, namely that S_(3,4), though not being a Lie subalgebra of the Lie algebra se(3) of the special Euclidean group SE(3), presents similar characteristics, which underlie the operation of a CV coupling. First, S_(3,4) is closed under two consecutive Lie bracket operations (or a Lie triple product), thus being referred to as a Lie triple system; second, taking the exponential of all twists in S_(3,4) generates a submanifold of SE (3), which is exactly the continuous-motion manifold of the coupling. In this paper, we first give a geometric characterization of the Lie product and the Lie triple product of a screw system. Then, we present a systematic identification of all Lie triple screw systems of se (3), by an approach based on both algebraic Lie group theory and descriptive screw theory.

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments. Mathematics Subject Classification (2000): 11E57 (primary); 22E40 (secondary). 1.1. Let us recall some notation and basic notions. Following [1] we will assume that n is even and n ≥ 4. The group of orientation preserving isometries of hyperbolic n-space is isomorphic to SO(1, n)o, the connected component of the identity of the special orthogonal group of signature (1, n), which can be identified with SO0(1, n), the subgroup of SO(1, n) preserving the upper half space. This group is not Zariski closed in SLn+1 thus in order to construct arithmetically defined subgroups of SO(1, n)o we consider arithmetic subgroups of the orthogonal group SO(1, n) or, more precisely, of groups G = SO( f ) where f is an admissible quadratic form defined over a totally real number field k (see [1, Section 2.1]). We have an exact sequence of k-isogenies: 1 → C → G φ → G → 1, (1.1) where G(k) Spin( f ) is the simply connected cover of G and C μ2 is the center of G. This induces an exact sequence in Galois cohomology (see [5, Section 2.2.3]) G(k) φ → G(k) δ → H1(k, C) → H1(k, G). (1.2) The main idea of this note is that by using (1.2) certain questions about arithmetic subgroups of G can be reduced to questions about the Galois cohomology group H1(k, C). A coherent collection of parahoric subgroups P = (Pv)v∈V f of G (V f = V f (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Received October 16, 2006; accepted in revised form March 12, 2007.

Given a field K equipped with a set of discrete valuations V , we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion K -algebra Q to quadratic forms over the function field K(Q) obtained via Morita equivalence. Using this we show that if (K,V) satisfies certain conditions, then the number of K -isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in V is finite and bounded by a value that depends on size of a quotient of the Picard group of V and the size of the kernel and cokernel of residue maps in Galois cohomology of K with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.