Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties.

1994 ◽  
Vol 1994 (453) ◽  
pp. 49-112 ◽  
Keyword(s):  
Hasse Principle ◽  
Weak Approximation

scholarly journals The Hardy–Littlewood conjecture and rational points

2014 ◽  
Vol 150 (12) ◽  
pp. 2095-2111 ◽  
Author(s):  
Yonatan Harpaz ◽  
Alexei N. Skorobogatov ◽  
Olivier Wittenberg
Keyword(s):  
Rational Points ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Ground Field ◽  
Formula One ◽  
Littlewood Conjecture ◽  
Pencils Of Conics

AbstractSchinzel’s Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer–Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is $\mathbb{Q}$ and the degenerate geometric fibres of the pencil are all defined over $\mathbb{Q}$, one can use this method to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy–Littlewood conjecture recently established by Green, Tao and Ziegler.

scholarly journals Hasse principle and weak approximation for multinorm equations

2014 ◽  
Vol 202 (1) ◽  
pp. 275-293 ◽  
Author(s):  
Cyril Demarche ◽  
Dasheng Wei
Keyword(s):  
Hasse Principle ◽  
Weak Approximation

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

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 Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

2014 ◽  
Vol 158 (1) ◽  
pp. 131-145 ◽  
Author(s):  
ARNE SMEETS
Keyword(s):  
Homogeneous Space ◽  
Number Field ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Generic Fibre ◽  
Principal Homogeneous Space ◽  
Smooth Compactification

AbstractLetkbe a number field andTak-torus. Consider a family of torsors underT, i.e. a morphismf:X→ ℙ1kfrom a projective, smoothk-varietyXto ℙ1k, the generic fibreXη→ η of which is a smooth compactification of a principal homogeneous space underT⊗kη. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation forX, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional ifk=Qand all non-split fibres offare defined overQ. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

scholarly journals Rational points on the intersection of three quadrics

2017 ◽  
Vol 13 (02) ◽  
pp. 273-289
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Complete Intersection ◽  
Arbitrary Number ◽  
Number Fields ◽  
Rational Points ◽  
Hasse Principle ◽  
Weak Approximation

We prove the Hasse principle and weak approximation for varieties defined by the smooth complete intersection of three quadratics in at least 19 variables, over arbitrary number fields.

Thue equations that simultaneously fail the Hasse principle

2021 ◽  
pp. 1-10
Author(s):  
PALOMA BENGOECHEA
Keyword(s):  
Positive Integer ◽  
Hasse Principle ◽  
Fixed Degree ◽  
Binary Forms ◽  
Absolute Value ◽  
Positive Proportion ◽  
The Absolute

Abstract We refine a previous construction by Akhtari and Bhargava so that, for every positive integer m, we obtain a positive proportion of Thue equations F(x, y) = h that fail the integral Hasse principle simultaneously for every positive integer h less than m. The binary forms F have fixed degree ≥ 3 and are ordered by the absolute value of the maximum of the coefficients.

scholarly journals Explicit methods for the Hasse norm principle and applications to A n and S n extensions

2021 ◽  
pp. 1-41
Author(s):  
ANDRÉ MACEDO ◽  
RACHEL NEWTON
Keyword(s):  
Galois Group ◽  
Computational Methods ◽  
Number Fields ◽  
Normal Closure ◽  
Explicit Methods ◽  
Weak Approximation ◽  
Norm Principle ◽  
Theoretical Results

Abstract Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

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