A local-global principle for quadratic forms

1975 ◽  
Vol 142 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Alexander Prestel
Keyword(s):  
Quadratic Forms ◽  
Global Principle

scholarly journals Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

2009 ◽  
Vol 145 (2) ◽  
pp. 309-363 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Fei Xu
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Integral Points ◽  
Integral Quadratic Form ◽  
Linear Algebraic Groups ◽  
Global Principle

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

scholarly journals A local global principle for quadratic forms over polynomial rings

1982 ◽  
Vol 74 (1) ◽  
pp. 264-269 ◽  
Author(s):  
Raman Parimala ◽  
R Sridharan
Keyword(s):  
Quadratic Forms ◽  
Polynomial Rings ◽  
Global Principle

scholarly journals Patching over Berkovich curves and quadratic forms

2019 ◽  
Vol 155 (12) ◽  
pp. 2399-2438
Author(s):  
Vlerë Mehmeti
Keyword(s):  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Analytic Geometry ◽  
Central Simple Algebras ◽  
Local Isotropy ◽  
Simple Algebras ◽  
Central Simple ◽  
Global Principle ◽  
Analytic Curves ◽  
Invent Math

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local–global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the $u$-invariant. The patching method we adapt was introduced by Harbater and Hartmann [Patching over fields, Israel J. Math. 176 (2010), 61–107] and further developed by these two authors and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263]. The results presented in this paper generalize those of Harbater, Hartmann, and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263] on the local–global principle and quadratic forms.

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.

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