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The proportion of genus one curves over $ℚ$ defined by a binary quartic that everywhere locally have a point

Manjul Bhargava

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

E-mail Address: bhargava@math.princeton.edu

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John Cremona

https://orcid.org/0000-0002-7212-0162

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

E-mail Address: j.e.cremona@warwick.ac.uk

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, and

Tom Fisher

University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

E-mail Address: T.A.Fisher@dpmms.cam.ac.uk

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https://doi.org/10.1142/S1793042121500147Cited by:4 (Source: Crossref)

Abstract

We consider the proportion of genus one curves over $ℚ$ of the form $z^{2} = f (x, y)$ where $f (x, y) \in ℤ [x, y]$ is a binary quartic form (or more generally of the form $z^{2} + h (x, y) z = f (x, y)$ where also $h (x, y) \in ℤ [x, y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree- $9$ rational function of $p$ for all odd $p$ (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

Keywords:

AMSC: 11G07, 11G05

References

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Vol. 17, No. 04

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History

Received 5 June 2020

Revised 17 July 2020

Accepted 17 July 2020

Published: 9 October 2020

Information

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Abstract

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The proportion of genus one curves over $ℚ$ defined by a binary quartic that everywhere locally have a point