Cohomology and the Brauer groups of diagonal surfaces

Gvirtz, Damián; Skorobogatov, Alexei N.

doi:10.1215/00127094-2021-0029

Cited by 4 publications

(9 citation statements)

References 39 publications

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“…We will build on previous work by Pham [35], Looijenga [30], Weil [46] and Ulmer [43]. The approach closely follows a framework developed by Gvirtz-Skorobogatov [20].…”

Section: Cohomology Of Weighted Diagonal Hypersurfacesmentioning

confidence: 99%

“…is not surjective. In order to decide which elements lie in the image, we make use of the cohomological machinery developed in [5,20], which in general applies to any surface over a number field with torsion-free geometric Picard group.…”

Section: Part 3 Proof Of the Main Theoremsmentioning

confidence: 99%

“…where A = (A 1 , A 2 , A 3 ) ∈ Z 3 . In this paper we capitalise on recent breakthroughs on the calculation of the Brauer groups of K3 surfaces by Corn-Nakahara [6] and Gvirtz-Skorobogatov [20], and calculate the complete Brauer group, including the transcendental part, for most of the surfaces in this family. We then study the number of such surfaces for which there is a Brauer-Manin obstruction to the Hasse principle, as the coefficients lie in a box |A i | ≤ T with T → ∞.…”

Section: Introductionmentioning

confidence: 99%

“…These are classes that do not lie in the algebraic Brauer group Br 1 X = ker(Br X → Br XQ). Traditionally, the transcendental part of the Brauer group has been the least understood, but recent methods developed for diagonal quartic surfaces [20] make it accessible in the case we consider. We adapt those methods for diagonal surfaces in weighted projective space to prove the following result, which shows that almost always the transcendental part of the Brauer group is trivial.…”

Section: Introductionmentioning

confidence: 99%

“…In Part 2 we calculate the transcendental Brauer groups of the surfaces X A via a spectral sequence framework due to Colliot-Thélène, Skorobogatov and the first author [5,20]. As a necessary prerequisite, we completely determine the integraladic middle cohomology of smooth projective weighted diagonal hypersurfaces using techniques from Hodge theory and the equivariant -adic Lefschetz trace formula.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative arithmetic of diagonal degree 2 K3 surfaces

2021

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In this paper we study the existence of rational points for the family of K3 surfaces over Q given byWhen the coefficients are ordered by height, we show that the Brauer group is almost always trivial, and find the exact order of magnitude of surfaces for which there is a Brauer-Manin obstruction to the Hasse principle. Our results show definitively that K3 surfaces can have a Brauer-Manin obstruction to the Hasse principle that is only explained by odd order torsion.

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“…We will build on previous work by Pham [35], Looijenga [30], Weil [46] and Ulmer [43]. The approach closely follows a framework developed by Gvirtz-Skorobogatov [20].…”

Section: Cohomology Of Weighted Diagonal Hypersurfacesmentioning

confidence: 99%

Section: Part 3 Proof Of the Main Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantitative arithmetic of diagonal degree 2 K3 surfaces

2021

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Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

Ieronymou

2023

Trans. Amer. Math. Soc.

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We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, under which the odd torsion does not obstruct weak approximation. We also note a systematic way to produce K 3 K3 surfaces over Q 2 \mathbb {Q}_2 with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.

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Diagonal quartic surfaces with a Brauer–Manin obstruction

Santens¹

2023

Compositio Math.

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In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

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Cohomology and the Brauer groups of diagonal surfaces

Abstract: We present a method for calculating the Brauer group of a surface given by a diagonal equation in projective space. For diagonal quartic surfaces with coefficients in Q we determine the Brauer groups over Q and Q(i).

Cited by 4 publications

References 39 publications

Quantitative arithmetic of diagonal degree 2 K3 surfaces

Quantitative arithmetic of diagonal degree 2 K3 surfaces

Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

Diagonal quartic surfaces with a Brauer–Manin obstruction

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