Quadratic forms ‘à la’ local 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.

is. The study of partially ordered sets is a kind of abstract mathematics of the most general sort, compared with the theory of algebraic integers which gave rise to ideal theory. I should like to say a few words about the distinction which is often made between abstract mathematics and the other kind, which may be called classical mathematics. Classical mathematics is based on the real number system, or equivalently on the system of positive integers or on the complex number system. Abstract mathematics is based on set theory. It studies the consequences of quite arbitrary postulate systems. From the point of view of the abstract mathematician, the real number system is a particular ordered field, or a particular locally compact topological ring; in other words, one of a class of similar abstract systems, and not necessarily more interesting than the others. From this same point of view, classical mathematics is a special case of a more general abstract theory. Actually, all mathematics is equally abstract. The real number system is no more concrete than, say, the theory of groups. Neither deals with the physical universe. To be sure, classical mathematics has seemed to be the kind most useful in applications to the sciences. However, perhaps this is an illusion, due to the time lag between the development of a theory and its application. The classical mathematics of the last century is being applied now. The abstract mathematics now being developed may turn out to be just as useful in the next century. Perhaps it will be found that you can kill just as many people just as efficiently using the theory of quasi-groups or p-adic fields as you can with the real number system. Of course many mathematicians will disagree with this view. But it is true now, just as it has been found to be true in the past, that mathematical results arrived at with no thought of their possible use are being applied to situations not imagined by their originators.

Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.

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.

A first characterization of the isomorphism classes of k -involutions for any reductive algebraic group defined over a perfect field was given in On the Classification of k -involutions I using 3 invariants.In Involutions of SL(n, k) , (n > 2) a full classification of all k -involutions on SL(n, k) for k algebraically closed, the real numbers, the p -adic numbers or a finite field was provided.In Isomorphy Classes of Involutions of SP(2n, k) , (n > 2) , a similar classification was given for all k -involutions of SP(2n, k) .In this paper, we find analogous results to develop a detailed characterization of the k -involutions of SO(n, k, ) , where is any non-degenerate symmetric bilinear form and k is any field not of characteristic 2 .We use these results to characterize the isomorphy classes of k -involutions of SO(n, k, ) for all bilinear forms, when char(k) = 2, 3 , and for some bilinear forms when char(k) = 3 .When n = 3, 4, 6, 8 , then the characterization considers all involutions.If n = 3, 4, 6, 8 , then the characterization only considers inner involutions.

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.

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.

A first characterization of the isomorphism classes of \(k\)-involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of \(\operatorname{SL}(n, k)\) with \(k\) any field of characteristic not equal to \(2\). We classify the isomorphism classes of involutions for \(k\) algebraically closed, the real numbers, the \(\mathfrak{p}\)-adic numbers and finite fields. We also determine in which cases the corresponding fixed point group \(H\) is \(k\)-anisotropic. In those cases the corresponding symmetric \(k\)-variety consists of semisimple elements.

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 investigate the ramification of embeddings of local lattices or quadratic forms over the ring of -adic numbers. It is proved that every primitive embedding decomposes uniquely into an orthogonal sum of minimal indecomposable embeddings, and all such embeddings are constructed for -elementary lattices. Ramification theory enables us to find the number of orbits of representations for forms and, in particular, for numbers by other quadratic forms over , and to calculate the local multipliers in the weight formula for representations of a form by a genus of quadratic forms.

As we saw in the last chapter, it is possible to obtain some non-trivial necessary conditions for the existence of orthogonal designs just by considering the equations that the coefficient matrices of an orthogonal design must satisfy. The ad-hoc procedures we give in the last chapter can, with difficulty, be pursued further. However, these procedures quickly become inadequate. To properly describe the solution to the “algebraic problem of orthogonal designs”, it is necessary to discuss the theory of quadratic and bilinear forms. Only then can the “algebraic problem” be put in proper perspective.

This book grew out of seminar held at the University of Paris 7 during the academic year 1985-86. The aim of the seminar was to give an exposition of the theory of the Metaplectic Representation (or Weil Representation) over a p-adic field. The book begins with the algebraic theory of symplectic and unitary spaces and a general presentation of metaplectic representations. It continues with exposes on the recent work of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture in the unramified case, representations of low rank). These lecture notes contain several original results. The book assumes some background in geometry and arithmetic (symplectic forms, quadratic forms, reductive groups, etc.), and with the theory of reductive groups over a p-adic field. It is written for researchers in p-adic reductive groups, including number theorists with an interest in the role played by the Weil Representation and -series in the theory of automorphic forms.

Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.