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

Author(s): Bhargava, Manjul; Cremona, John; Fisher, Tom

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Abstract: We show that the proportion of plane cubic curves over Q p that have a Q p -rational point is a rational function in p , where the rational function is independent of p ,andwe determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over Q that have points everywhere locally; numerically, this density is shown to be ≈ 97 . 3%.
Publication Date: Jun-2016
Electronic Publication Date: 5-Oct-2015
Citation: Bhargava, Manjul, Cremona, John, Fisher, Tom. (2016). The proportion of plane cubic curves over Q that everywhere locally have a point. INTERNATIONAL JOURNAL OF NUMBER THEORY, 12 (1077 - 1092. doi:10.1142/S1793042116500664
DOI: doi:10.1142/S1793042116500664
ISSN: 1793-0421
EISSN: 1793-7310
Pages: 1077 - 1092
Type of Material: Journal Article
Journal/Proceeding Title: INTERNATIONAL JOURNAL OF NUMBER THEORY
Version: Author's manuscript



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