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The Hasse principle for lines on diagonal surfaces

Published online by Cambridge University Press:  30 October 2015

JÖRG JAHNEL  and
DANIEL LOUGHRAN
JÖRG JAHNEL
Affiliation:
Department Mathematik, Walter-Flex-Strasse 3, Universität Siegen, D-57072, Siegen, Germany. e-mail: jahnel@mathematik.uni-siegen.de
DANIEL LOUGHRAN
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany. e-mail: loughran@math.uni-hannover.de
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Abstract

Given a number field k and a positive integer d, in this paper we consider the following question: does there exist a smooth diagonal surface of degree d in $\mathbb{P}$3 over k which contains a line over every completion of k, yet no line over k? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erdős.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 107 - 119
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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