The Hasse principle for pairs of quadratic forms

1982 ◽  
pp. 237-246 ◽  
Author(s):  
D.F. Coray
Keyword(s):  
Quadratic Forms ◽  
Hasse Principle

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.

scholarly journals On Hasse principle for division of quadratic forms

1979 ◽  
Vol 31 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Takashi ONO ◽  
Hiroyuki YAMAGUCHI
Keyword(s):  
Quadratic Forms ◽  
Hasse Principle

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.

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