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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
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 | ||||||||||
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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: |
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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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