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Research in Number Theory

, 4:33 | Cite as

Brauer–Manin obstructions on degree 2 K3 surfaces

  • Patrick Corn
  • Masahiro Nakahara
Research
  • 32 Downloads

Abstract

We analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over \({{\mathbb {Q}}}\) given by double covers of \({{\mathbb {P}}^{2}}\) ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer–Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.

Keywords

Rational points Hasse principle Brauer–Manin obstruction 

Mathematics Subject Classification

14G05 11G35 14F22 

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK

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