Odd order Brauer–Manin obstruction on diagonal quartic surfaces

Ieronymou, Evis; Skorobogatov, Alexei N.

doi:10.1016/j.aim.2014.11.004

Cited by 35 publications

(47 citation statements)

References 14 publications

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“…where Γ acts trivially on Z n . Now (i) follows from the exact sequence (8). Part (ii) follows from Proposition 4.1, so it remains to establish part (iii).…”

Section: Kummer Varieties Attached To Products Of Abelian Varietiesmentioning

confidence: 94%

Kummer varieties and their Brauer groups

Skorobogatov

Zarhin

2017

Pure and Applied Mathematics Quarterly

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We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the nonemptyness of the Brauer-Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.

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“…where Γ acts trivially on Z n . Now (i) follows from the exact sequence (8). Part (ii) follows from Proposition 4.1, so it remains to establish part (iii).…”

Section: Kummer Varieties Attached To Products Of Abelian Varietiesmentioning

confidence: 94%

Kummer varieties and their Brauer groups

Skorobogatov

Zarhin

2017

Pure and Applied Mathematics Quarterly

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“…Another example of a rationally connected variety X over a number field k such that X(A k ) Br(X) = X(A k ) Br 1 (X) is given in [DLAN17]. Transcendental elements and their influence on the Brauer-Manin set have received a lot of attention for other classes of varieties as well (see [Wit04], [Ier10], [HVAV11], [Pre13], [HVA13], [IS15], [New16], [CV15], [MSTVA16] for examples of K3 and Enriques surfaces for which transcendental elements play a role in the Brauer-Manin set).…”

Section: Over Number Fields: General Contextmentioning

confidence: 99%

“…The analogous problem for X(A Q ) Br 1 (X) was solved by Bright [Bri06]. It is known that transcendental elements cannot be ignored in this context: there exist diagonal quartic surfaces X over Q such that X(A Q ) Br(X) = X(A Q ) Br 1 (X) (see [Ier10], [Pre13], [IS15]). There also exist K3 surfaces X over Q such that…”

Section: K3 Surfacesmentioning

confidence: 99%

Rational points and zero-cycles on rationally connected varieties over number fields

Wittenberg¹

2018

Proceedings of Symposia in Pure Mathematics

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We report on progress in the qualitative study of rational points on rationally connected varieties over number fields, also examining integral points, zero-cycles, and non-rationally connected varieties. One of the main objectives is to highlight and explain the many recent interactions with analytic number theory. p. 174]). Let X be a smooth, proper, geometrically irreducible variety defined over a number field k. If X is rationally connected, then X(k) is dense in X(A k ) Br(X) . Conjecture 2.3By "X is rationally connected", we mean that the variety X ⊗ k k is rationally connected in the sense of Campana, Kollár, Miyaoka and Mori, where k denotes an algebraic closure of k. Equivalently, for any algebraically closed field K containing k, two general K-points of X can be joined by a rational curve defined over K. We recall that examples of rationally connected varieties include geometrically unirational varieties (trivial), Fano varieties (see [Cam92], [KMM92]), and fibrations into rationally connected varieties over a rationally connected base (see [GHS03]).Conjecture 2.3 is wide open even for surfaces.Remarks 2.4. (i) The group Br 0 (X) = Im(Br(k) → Br(X)) does not contribute to the Brauer-Manin set: the global reciprocity law implies the equality X(A k ) B = X(A k ) B+Br 0 (X) for any subgroup B ⊆ Br(X). When Br(X)/Br 0 (X) is finite, the subset X(A k ) Br(X) ⊆ X(A k ) is therefore cut out by finitely many conditions. In particular, in this case, it is an open subset of X(A k ).(ii) When X is rationally connected, or, more generally, when X is simply connected and H 2 (X, O X ) = 0, an analysis of the Hochschild-Serre spectral sequence and of the Brauer group of X ⊗ k k implies that Br(X)/Br 0 (X) is finite (see [CTS13b, Lemma 1.1]). Presumably, this quotient should be finite under the sole assumption that X is simply connected, but this is out of reach of current knowledge. (As follows from [CTS13a] and [CTS13b, §4], the finiteness of the ℓ-primary torsion subgroup of Br(X)/Br 0 (X) is equivalent, when X is simply connected, to the ℓ-adic Tate conjecture for divisors on X.) (iii) Letting v be any archimedean place of k and noting that the image of the projection map X(A k ) Br(X) → X(k v ) is a union of connected components, we see

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“…It should be noted that Ieronymou and Skorobogatov [23] have shown that odd-order elements in the Brauer groups of diagonal quartic surfaces over Q, when they do exist, can never obstruct the Hasse principle. They do this by calculating that, if A has order p = 3 or 5 in Br X/ Br Q, then the corresponding evaluation map A : X(Q p ) → Br Q p [p] is surjective.…”

Section: Vanishing Of Brauer-manin Obstructionsmentioning

confidence: 99%

Bad reduction of the Brauer-Manin obstruction

Bright

2015

Journal of the London Mathematical Society

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We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem inétale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how this surjectivity can be used to prove vanishing of the Brauer-Manin obstruction on varieties over number fields.

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Odd order Brauer–Manin obstruction on diagonal quartic surfaces

Abstract: We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.

Cited by 35 publications

References 14 publications

Kummer varieties and their Brauer groups

Kummer varieties and their Brauer groups

Rational points and zero-cycles on rationally connected varieties over number fields

Bad reduction of the Brauer-Manin obstruction

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