Quantitative arithmetic of diagonal degree 2 K3 surfaces

Gvirtz, D. , Loughran, D. and Nakahara, M. (2021) Quantitative arithmetic of diagonal degree 2 K3 surfaces. Mathematische Annalen, 384(1-2), pp. 1-75. (doi: 10.1007/s00208-021-02280-w)

Abstract

In this paper we study the existence of rational points for the family of K3 surfaces over Q given by w2=A1x61+A2x62+A3x63. 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.

• Item Type: Articles
• Additional Information: 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.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gvirtz, Dr Damian
• Authors: Gvirtz, D., Loughran, D., and Nakahara, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Mathematische Annalen
• Publisher: Springer
• ISSN: 0025-5831
• ISSN (Online): 1432-1807
• Published Online: 11 October 2021
• Copyright Holders: Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
• First Published: First published in Mathematische Annalen 384(1-2): 1-75
• Publisher Policy: Reproduced in accordance with the publisher copyright policy