The Brauer group of torsors and its arithmetic applications

Harari, David; Skorobogatov, Alexei N.

doi:10.5802/aif.1998

Cited by 6 publications

(3 citation statements)

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“…It follows that the homomorphism Br(X) → Br(X ′ ) is injective. Now a diagram chase in diagram(23) proves the exactness of the second row in that diagram. This completes the proof of the exactness of the top row of diagram(8).…”

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confidence: 83%

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Manin obstruction to strong approximation for homogeneous spaces

Borovoi¹,

Demarche²

2013

Comment. Math. Helv.

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For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group Br(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.

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confidence: 83%

“…We cannot describe easily the image of this map in general. However, Harari and Skorobogatov studied this map in particular cases (universal torsors for instance): see [23], Theorems 1.6 and 1.7.…”

Section: And Thementioning

confidence: 99%

Manin obstruction to strong approximation for homogeneous spaces

Borovoi¹,

Demarche²

2013

Comment. Math. Helv.

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“…Here we consider a projective, surjective morphism p : X -t pI with smooth generic fibre XTj' Assume also that XTj has a k(1])-point (this technical condition is satisfied in most applications, e.g., if XTj is geometrically rationally connected, by a recent result of Graber, Harris and Starr [15] Here aga in it is possible to replace "geomet rica lly integral" by "split" in t he t hird cond ition (4 It is also possible to combine open descent with t he fibration method to obtain generalizati ons of Th eorem 3.4.1 when at most 2 (or 3 in very special cases) fibres are degenerate (see [21]). …”

Section: 2)mentioning

confidence: 99%

Weak Approximation on Algebraic Varieties

Harari¹

2004

Progress in Mathematics

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Comparing descent obstruction and Brauer-Manin obstruction for open varieties

Cao

Demarche

2019

Trans. Amer. Math. Soc.

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We provide a relation between Brauer-Manin obstruction and descent obstruction for torsors over not necessarily proper varieties under a connected linear algebraic group or a group of multiplicative type. Such a relation is also refined for torsors under a torus. The equivalence between descent obstruction andétale Brauer-Manin obstruction for smooth projective varieties is extended to smooth quasi-projective varieties, which provides the perspective to study integral points. π * − → Br(X)) where X π − → Spec(k) is the structure morphism, and Br a (X) = Br 1 (X)/Br 0 (X). For any subgroup B of Br(X), one can define the Brauer-Manin setwith respect to B. When B = Br(X), we simply write this Brauer-Manin set as X(A k ) Br .

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The Brauer group of torsors and its arithmetic applications

Cited by 6 publications

References 2 publications

Manin obstruction to strong approximation for homogeneous spaces

Manin obstruction to strong approximation for homogeneous spaces

Weak Approximation on Algebraic Varieties

Comparing descent obstruction and Brauer-Manin obstruction for open varieties

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