Quadratic Forms

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Forms ◽  
Splitting Field ◽  
Quadratic Space ◽  
Polar Spaces ◽  
Quadratic Module ◽  
Quadratic Modules ◽  
Hyperbolic Planes ◽  
Scalar Extensions

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

1966 ◽  
Vol 18 ◽  
pp. 920-942 ◽  
Author(s):  
Allan Trojan
Keyword(s):  
Vector Space ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
The Other ◽  
Quadratic Space ◽  
Integral Extension ◽  
Ring Of Integers ◽  
Extension Of Isometries ◽  
Necessary And Sufficient

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

scholarly journals Representations of quadratic forms

1978 ◽  
Vol 69 ◽  
pp. 117-120
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Direct Summand ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
List Type

We have shown in [1]Theorem A. Let L be a lattice in a regular quadratic space U over Q; then L has a submodule M satisfying the following conditions 1),2): 1)dM ≠ 0, rank M = rank L — 1, and M is a direct summand of L as a module.2)Let L′ be a lattice in some regular quadratic space U′ over Q satisfying dL′ = dL, rank L′ — rank L, tp(L′) ≥ tp(L) for any prime p. If there is an isometry α from M into L′ such that α(M) is a direct summand of L′ as a module, then L′ is isometric to L.

Quadratic Forms of Type F4

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
Scalar Multiplication ◽  
Quadratic Space

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

A Note on Quadratic forms Over Arbitrary Semi-Local Rings

1975 ◽  
Vol 27 (3) ◽  
pp. 513-527
Author(s):  
K. I. Mandelberg
Keyword(s):  
Commutative Ring ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Local Rings ◽  
Quadratic Mapping ◽  
Finitely Generated ◽  
Bilinear Mapping ◽  
Natural Mapping

Let R be a commutative ring. A bilinear space (E, B) over R is â finitely generated projective R-module E together with a symmetric bilinear mapping B:E X E →R which is nondegenerate (i.e. the natural mapping E → HomR(E﹜ R) induced by B is an isomorphism). A quadratic space (E, B, ) is a bilinear space (E, B) together with a quadratic mapping ϕ:E →R such that B(x, y) = ϕ (x + y) — ϕ (x) — ϕ (y) and ϕ (rx) = r2ϕ (x) for all x, y in E and r in R. If 2 is a unit in R, then ϕ (x) = ½. B﹛x,x) and the two types of spaces are in obvious 1 — 1 correspondence.

Collineations of Polar Spaces

1985 ◽  
Vol 37 (2) ◽  
pp. 296-309 ◽  
Author(s):  
Donald G. James
Keyword(s):  
Projective Space ◽  
Quadratic Forms ◽  
Fundamental Theorem ◽  
Valuation Ring ◽  
Finite Dimension ◽  
Vector Spaces ◽  
Polar Spaces ◽  
Hermitian Forms ◽  
Bijective Mapping ◽  
Integral Structure

The fundamental theorem of projective geometry describes the bijective collineations between two projective spaces PV and PV′ of finite dimension (greater than one) over division rings k and k′ in terms of an isomorphism φ:k → k′ and a φ-semilinear bijective mapping between the underlying vector spaces V and V′. Tits [9, Theorem 8.611] has given an extensive generalization of this theorem to embeddable polar spaces induced by polarities coming from either (σ, )-hermitian forms or from (σ, )-quadratic forms with Witt indices at least two. In another direction, Klingenberg [7] and later André [1] and Rado [8], have generalized the fundamental theorem by considering non-injective collineations. Now the isomorphism φ must be replaced by a place φ:k → k′ ∪ ∞ and an integral structure over the valuation ring A = φminus1(k′) is induced into the projective space PV.

Quadratic Forms of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
The Other ◽  
Quadratic Space ◽  
Division Algebras ◽  
Arbitrary Characteristic ◽  
Definition Of ◽  
Quaternion Division Algebra

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

scholarly journals Zeros of pairs of quadratic forms

2018 ◽  
Vol 2018 (739) ◽  
pp. 41-80
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Quadratic Forms ◽  
Number Fields ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Local Problem ◽  
Hyperbolic Planes

Abstract We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in eight variables. The argument develops work of Colliot-Thélène, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off three hyperbolic planes.

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