The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point

Manjul Bhargava; John Cremona; Tom Fisher

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arXiv · 1311.5578

The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point

Manjul Bhargava·John Cremona·Tom Fisher

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Abstract

We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-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 ${\mathbb Q}$ that have points everywhere locally; numerically, this density is shown to be $\approx 97.3\%$.

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Manjul Bhargava, John Cremona, Tom Fisher. 2013-11-21. The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point. https://arxiv.org/abs/1311.5578

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