Pathologies of the Brauer–Manin obstruction

Colliot-Thélène, Jean-Louis; Pál, Ambrus; Skorobogatov, Alexei N.

doi:10.1007/s00209-015-1565-x

Cited by 22 publications

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“…The fundamental reason is the Harpaz-Schlank theorem quoted above which implies that if HSP holds for X then the Brauer-Manin obstruction applied toétale covers is the only obstruction for the Hasse principle. Since now there are many counter-examples to this claim (see [28], [20] and [3]) we get that there are two-and three-dimensional counterexamples to HSP. However we can offer some positive results; see Theorems 13.3, 13.7 and 14.8 in the next two sections.…”

Section: The Homotopy Section Principlementioning

confidence: 88%

Étale homotopy equivalence of rational points on algebraic varieties

Pál

2015

Algebra Number Theory

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It is possible to talk about theétale homotopy equivalence of rational points on algebraic varieties by using a relative version of theétale homotopy type. We show that over p-adic fields rational points are homotopy equivalent in this sense if and only if they areétale-Brauer equivalent. We also show that over the real field rational points on projective varieties areétale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Châtelet surfaces.

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Section: The Homotopy Section Principlementioning

confidence: 88%

Étale homotopy equivalence of rational points on algebraic varieties

Pál

2015

Algebra Number Theory

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“…In §6 we give an example of a minimal conic bundle over an elliptic curve for which degrees do not capture the Brauer-Manin obstruction. This is unsurprising (and somewhat less disappointing) given the known pathologies of the Brauer-Manin obstruction on quadric fibrations [CTPS16]. In §6 we note that degrees do capture the Brauer-Manin obstruction on minimal rational conic bundles and on Severi-Brauer bundles over elliptic curves with finite Tate-Shafarevich group (See Theorem 6.1).…”

Section: Introductionmentioning

confidence: 95%

Degree and the Brauer–Manin obstruction

Creutz

Viray

2018

Alg. Number Th.

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Let X ⊂ P n k be a smooth projective variety of degree d over a number field k and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. In the case of Kummer surfaces we show, more specifically, that the obstruction is already given by the 2-primary torsion, and indeed that this holds for higher-dimensional Kummer varieties as well. We construct a conic bundle over an elliptic curve that shows that, in general, the answer is no.Question 1.1. Suppose that X ֒→ P n is embedded as a subvariety of degree d in projective space. Does the d-primary subgroup of Br X capture the Brauer-Manin obstruction to rational points on X?More intrinsically, let us say that degrees capture the Brauer-Manin obstruction on X if the d-primary subgroup of Br X captures the Brauer-Manin obstruction to rational points on X for all integers d that are the degree of some k-rational globally generated ample line bundle on X. Since any such line bundle determines a degree d morphism to projective space and conversely, it is clear that the answer to Question 1.1 is affirmative when degrees capture the Brauer-Manin obstruction.1.1. Summary of results. In general, the answer to Question 1.1 can be no (see the discussion in §1.2). However, there are many interesting classes of varieties for which the answer is yes. We prove that degrees capture the Brauer-Manin obstruction for torsors under abelian varieties, for Kummer surfaces, and, assuming finiteness of Tate-Shafarevich groups of elliptic curves, for bielliptic surfaces. We also deduce (from various results appearing in the literature) that degrees capture the Brauer-Manin obstruction for all geometrically rational minimal surfaces.Assuming finiteness of Tate-Shafarevich groups, one can deduce the result for torsors under abelian varieties rather easily from a theorem of Manin (see Remark 4.4 and Proposition 4.9). In §4 we unconditionally prove the following much stronger result.Theorem 1.2. Let X be a k-torsor under an abelian variety, let B ⊂ Br X be any subgroup, and let d be any multiple of the period of X. In particular, d could be taken to be the degree of a k-rational globally generated ample line bundle.This not only shows that degrees capture the Brauer-Manin obstruction (apply the theorem with B = Br X), but also that the Brauer classes with order relatively prime to d cannot provide any obstructions to the existence of rational points. Remark 1.3. As one ranges over all torsors of period d under all abelian varieties over number fields, elements of arbitrarily large order in of (Br V )[d ∞ ] are required to produce the obstruction (see Proposition 4.10).We use Theorem 1.2 to ded...

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“…Hence X p is everywhere locally soluble. Furthermore assume p ≡ 1 (mod 8) to ensure that the invariant maps over Q 2 will carry out same as in the proof. Then X p will be a counterexample to the Hasse principle given by a 3-torsion Brauer class in the algebraic Brauer group.…”

Section: P =mentioning

confidence: 99%

“…This is a contradiction, so we may assume that y 2 and z 2 are both odd. Now we use formula (8) to compute the invariant. We get…”

Section: Proof Of (C)mentioning

confidence: 99%

“…In 1971, Manin [32] identified an obstruction to the existence of rational points using the group Br(X) := H 2 et (X, G m ) tors , now known as the Brauer-Manin obstruction, which he used successfully to explain some of the counterexamples to the Hasse principle that were known at the time. Since then, groundbreaking work of Skorobogatov [25] has shown that the Brauer-Manin obstruction is not the only one for general varieties over number fields (see also [2,8,22,29]) but ColliotThélène conjectured [6] that the Brauer-Manin obstruction is the only obstruction to rational points for smooth, projective, geometrically rationally connected varieties; that is, if X is such a variety with points everywhere locally, and there is no Brauer-Manin obstruction to rational points on X, then X should have a rational point.…”

Section: Introductionmentioning

confidence: 99%

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Brauer–Manin obstructions on degree 2 K3 surfaces

Corn

Nakahara

2018

Res. number theory

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We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P 2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

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Pathologies of the Brauer–Manin obstruction

Cited by 22 publications

References 22 publications

Étale homotopy equivalence of rational points on algebraic varieties

Étale homotopy equivalence of rational points on algebraic varieties

Degree and the Brauer–Manin obstruction

Brauer–Manin obstructions on degree 2 K3 surfaces

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