Bhargava, Manjul, Cremona, J. E. and Fisher, Tom (2016) The proportion of plane cubic curves over Q that everywhere locally have a point. International Journal of Number Theory, 12 (4). 1077. doi:10.1142/S1793042116500664

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Abstract

We show that the proportion of plane cubic curves over Qpℚp that have a Qpℚp-rational point is a rational function in pp, where the rational function is independent of pp, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over Qℚ that have points everywhere locally; numerically, this density is shown to be ≈97.3%≈97.3%.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Curves, Plane, Equations, Cubic
Journal or Publication Title:	International Journal of Number Theory
Publisher:	World Scientific Publishing Co. Pte. Ltd.
ISSN:	1793-0421
Official Date:	June 2016
Dates:	Date Event June 2016 Published 5 October 2015 Available 19 July 2015 Accepted 24 April 2015 Submitted
Volume:	12
Number:	4
Article Number:	1077
DOI:	10.1142/S1793042116500664
Status:	Peer Reviewed
Publication Status:	Published
Funder:	Simons Foundation (SF), National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	DMS-1001828 (NSF), Grant EP/K034383/1 (EPSRC)
Related URLs:	Publisher

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