Quantitative arithmetic of diagonal degree 2 K3 surfaces

Damián Gvirtz, Daniel Loughran, Masahiro Nakahara

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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 languageEnglish
Pages (from-to)1-75
JournalMathematische Annalen
Volume384
Early online date11 Oct 2021
DOIs
Publication statusPublished - 31 Oct 2022

Bibliographical 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 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.

ASJC Scopus subject areas

  • Mathematics(all)

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