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

Manjul Bhargava; John Cremona; Tom Fisher

doi:10.1142/S1793042116500664

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

Manjul Bhargava
, John Cremona
, Tom Fisher

Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we 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%.

Original language	English (US)
Pages (from-to)	1077-1092
Number of pages	16
Journal	International Journal of Number Theory
Volume	12
Issue number	4
DOIs	https://doi.org/10.1142/S1793042116500664
State	Published - Jun 1 2016

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Plane cubic curves
local solubility
random Diophantine equations

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10.1142/S1793042116500664

Cite this

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abstract = "We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we 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\%.",

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AU - Cremona, John

AU - Fisher, Tom

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N2 - We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we 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%.

AB - We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we 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%.

KW - Plane cubic curves

KW - local solubility

KW - random Diophantine equations

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

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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