Quantitative arithmetic of diagonal degree 2 K3 surfaces

Gvirtz, Damián; Loughran, Daniel; Nakahara, Masahiro

doi:10.1007/s00208-021-02280-w

Cited by 2 publications

(2 citation statements)

References 34 publications

(57 reference statements)

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“…However, the method used to resolve these issues also allows us, without too much extra effort, to find an asymptotic formula. This is a stronger type of result than the correct order of magnitude in [GLN22]. We hope that this method can also be applied to find asymptotic formulas for other families.…”

Section: Introductionmentioning

confidence: 88%

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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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Section: Introductionmentioning

confidence: 88%

“…Recently Gvirtz, Loughran and Nakahara [GLN22] investigated how often diagonal degree K3 surfaces have a Brauer–Manin obstruction and were able to obtain the correct order of magnitude. We instead look at another family of K3 surfaces, namely the diagonal quartic surfaces.…”

Section: Introductionmentioning

confidence: 99%

Diagonal quartic surfaces with a Brauer–Manin obstruction

Santens¹

2023

Compositio Math.

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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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Quantitative arithmetic of diagonal degree 2 K3 surfaces

Cited by 2 publications

References 34 publications

Diagonal quartic surfaces with a Brauer–Manin obstruction

Diagonal quartic surfaces with a Brauer–Manin obstruction

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

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