Special groups: Boolean-theoretic methods in the theory of quadratic forms

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.

Special groups are an axiomatization of the algebraic theory of quadratic forms over fields. It is known that any finite reduced special group is the special group of some field. We show that any special group that is the projective limit of a projective system of finite reduced special groups is also the special group of some field.

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.

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.

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

Field invariants under totally positive quadratic extensions

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed.

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

In an earlier paper we gave an approximation theory of focal points and focal intervals. The fundamental purpose of this paper is to show that the ideas and methods of that paper can be used to give an approximation theory for oscillation for linear selfadjoint differential (and integral-differential) equations. These methods follow from the theory of quadratic forms given by Hestenes. In §1 we give the preliminaries needed for this paper. In §2 we define oscillation for quadratic control problems and discuss their connection with differential equations. In §3 we give our main approximation results relating the oscillation points λ n ( σ 0 ) ( n = i , 2 , 3 , ⋯ ) {\lambda _n}({\sigma _0})(n = i,2,3, \cdots ) for a given problem to the oscillation points λ n ( σ ) ( n = 1 , 2 , 3 , ⋯ ) {\lambda _n}(\sigma )(n = 1,2,3, \cdots ) of an approximating problem. The element σ \sigma belongs to a metric space ( Σ , d ) (\Sigma ,d) . The main result is to show that the n n th oscillation point λ n ( σ ) {\lambda _n}(\sigma ) is a continuous function of σ ( n = i , 2 , 3 , ⋯ ) \sigma (n = i,2,3, \cdots ) . For completeness in §4 we present an example for fourth order differential equations where the approximation is by discrete problems. Thus oscillation points can be “easily” computed by numerical algorithms.

Let M and L be quadratic lattices over the maximal order of an algebraic number field. In case of dealing with representations of M by L, they sometimes assume certain indefiniteness and the condition rank L-rank M ≥ 3. In this case, representation problems are reduced not to global but to local problems by virtue of the strong approximation theorem for rotations and of the fact that for regular quadratic spaces U, V over a non-archimedian local field there is an isometry from U to V if dim V — dim U ≥ 3.

A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings H'F and GWF are isomorphic.Here H'F = H'(Ga\(Fq/F),Z/2Z)),where Fq is the quadratic closure of F , and GWF is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F .In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or H'K is 'close' to H'F .A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K.This relative theory extends and includes the previously developed absolute case where K = Fq .Applications are also given to show that H'F and GWF are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

There are connections, already seen by Gauss himself, between his analytic work and the theory of quadratic forms. Some fragments from the early 1800s show that Gauss was familiar with the rudiments of a theory which eventually found its completion in Klein’s and Fricke’s work on modular forms. Both F. Klein and R. Fricke were deeply involved in the publication of Gauss’s work and had the opportunity to study Gauss’s fragments closely.