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The proportion of plane cubic curves over Q that everywhere locally have a point

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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 ISSN 1793-0421.

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Official URL: http://dx.doi.org/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:
DateEvent
June 2016Published
5 October 2015Available
19 July 2015Accepted
24 April 2015Submitted
Volume: 12
Number: 4
Article Number: 1077
DOI: 10.1142/S1793042116500664
Status: Peer Reviewed
Publication Status: Published
Date of first compliant deposit: 1 February 2016
Date of first compliant Open Access: 5 October 2016
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)
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