Quantitative arithmetic of diagonal degree 2 K3 surfaces

Damián Gvirtz; Daniel Loughran; Masahiro Nakahara

doi:10.1007/s00208-021-02280-w

Quantitative arithmetic of diagonal degree 2 K3 surfaces

Damián Gvirtz, Daniel Loughran, Masahiro Nakahara

Department of Mathematical Sciences

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Abstract

In this paper we study the existence of rational points for the family of K3 surfaces over Q given by w2=A1x16+A2x26+A3x36.When 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.

Original language	English
Pages (from-to)	1-75
Journal	Mathematische Annalen
Volume	384
Early online date	11 Oct 2021
DOIs	https://doi.org/10.1007/s00208-021-02280-w
Publication status	Published - 31 Oct 2022

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00208-021-02280-w

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This is a post-peer-review, pre-copyedit version of an article published in Mathematische Annalen. The final authenticated version is available online at: https://doi.org/10.1007/s00208-021-02280-w
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title = "Quantitative arithmetic of diagonal degree 2 K3 surfaces",

abstract = "In this paper we study the existence of rational points for the family of K3 surfaces over Q given by w2=A1x16+A2x26+A3x36.When 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.",

author = "Dami{\'a}n Gvirtz and Daniel Loughran and Masahiro Nakahara",

note = "Funding Information: We are very grateful to Tim Browning for discussions on some of the analytic arguments and to Igor Dolgachev and Eduard Looijenga for discussions on their work regarding the cohomology of diagonal hypersurfaces. This work was partly undertaken at the Institut Henri Poincar{\'e} during the trimester “Reinventing rational points”; the authors thank the organisers and staff for ideal working conditions. The first-named author was supported by EP/L015234/1, the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory). Part 2 has overlaps with Chapters 11 and 12 of his PhD thesis. The second and third-named authors are supported by EPSRC grant EP/R021422/2. ",

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doi = "10.1007/s00208-021-02280-w",

language = "English",

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AU - Gvirtz, Damián

AU - Loughran, Daniel

AU - Nakahara, Masahiro

N1 - Funding Information: We are very grateful to Tim Browning for discussions on some of the analytic arguments and to Igor Dolgachev and Eduard Looijenga for discussions on their work regarding the cohomology of diagonal hypersurfaces. This work was partly undertaken at the Institut Henri Poincaré during the trimester “Reinventing rational points”; the authors thank the organisers and staff for ideal working conditions. The first-named author was supported by EP/L015234/1, the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory). Part 2 has overlaps with Chapters 11 and 12 of his PhD thesis. The second and third-named authors are supported by EPSRC grant EP/R021422/2.

PY - 2022/10/31

Y1 - 2022/10/31

N2 - In this paper we study the existence of rational points for the family of K3 surfaces over Q given by w2=A1x16+A2x26+A3x36.When 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.

AB - In this paper we study the existence of rational points for the family of K3 surfaces over Q given by w2=A1x16+A2x26+A3x36.When 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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JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

ER -

Quantitative arithmetic of diagonal degree 2 K3 surfaces

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Quantitative arithmetic geometry

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Quantitative arithmetic of diagonal degree 2 K3 surfaces

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Quantitative arithmetic geometry

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