Beyond the Manin obstruction

Skorobogatov, Alexei N.

doi:10.1007/s002220050291

Cited by 95 publications

(71 citation statements)

References 10 publications

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“…However, they developed their theory under the simplifying assumption that the varieties involved are proper. Skorobogatov developed a variant under less stringent assumptions in [22]. Descent on open varieties also features in [11] and [3].…”

Section: Universal Torsorsmentioning

confidence: 99%

Universal torsors and values of quadratic polynomials represented by norms

2014

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Let K /k be an extension of number fields, and let P(t) be a quadratic polynomial over k. Let X be the affine variety defined by P(t) = N K /k (z). We study the Hasse principle and weak approximation for X in three cases. For [K : k] = 4 and P(t) irreducible over k and split in K , we prove the Hasse principle and weak approximation. For k = Q with arbitrary K , we show that the BrauerManin obstruction to the Hasse principle and weak approximation is the only one. For [K : k] = 4 and P(t) irreducible over k, we determine the Brauer group of smooth proper models of X . In a case where it is non-trivial, we exhibit a counterexample to weak approximation.

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Section: Universal Torsorsmentioning

confidence: 99%

Universal torsors and values of quadratic polynomials represented by norms

2014

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“…We say that the Brauer-Manin obstruction is the only obstruction for the existence of zero cycles on X if It is known that the analogous Brauer-Manin obstruction for the existence of rational points is not the only obstruction to the existence of rational points on X (Skorobogatov [28]). …”

Section: R Parimalamentioning

confidence: 99%

A Hasse principle for quadratic forms over function fields

Parimala

2014

Bull. Amer. Math. Soc.

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Abstract. We describe the classical Hasse principle for the existence of nontrivial zeros for quadratic forms over number fields, namely, local zeros over all completions at places of the number field imply nontrivial zeros over the number field itself. We then go on to explain more general questions related to the Hasse principle for nontrivial zeros of quadratic forms over function fields, with reference to a set of discrete valuations of the field. This question has interesting consequences over function fields of p-adic curves. We also record some open questions related to the isotropy of quadratic forms over function fields of curves over number fields.

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“…Proof. The main theorem of descent theory of Colliot-Thélène and Sansuc [1987], as extended by Skorobogatov (see [Skorobogatov 1999] and [Skorobogatov 2001, Thm. 6…”

Section: Relation With the Brauer-manin Obstructionmentioning

confidence: 99%

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2007

ANT

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Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points on X within the adelic points. Our main result is that if a curve C/k maps nontrivially into an abelian variety A/k such that A(k) is finite and X(k, A) has no nontrivial divisible element, then the information coming from finite abelian descent cuts out precisely the rational points of C. We conjecture that this is the case for all curves of genus at least 2. We relate finite descent obstructions to the Brauer-Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by finite abelian descent. Our conjecture therefore implies that the Brauer-Manin obstruction against rational points is the only one on curves.

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Beyond the Manin obstruction

Cited by 95 publications

References 10 publications

Universal torsors and values of quadratic polynomials represented by norms

Universal torsors and values of quadratic polynomials represented by norms

A Hasse principle for quadratic forms over function fields

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