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

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Bharagava , Manjul , Cremona, J. E. and Fisher, Tom (2021) The proportion of genus one curves over Q defined by a binary quartic that everywhere locally have a point. International Journal of Number Theory, 17 (4). pp. 903-923. doi:10.1142/S1793042121500147 ISSN 1793-0421. [ 🗎 Public]. [ (✓) hoa:511 ]

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Official URL: https://doi.org/10.1142/S1793042121500147

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Abstract

We consider the proportion of genus one curves over Q of the form z2=f(x,y) where f(x,y)∈Z[x,y] is a binary quartic form (or more generally of the form z2+h(x,y)z=f(x,y) where also h(x,y)∈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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Diophantine equations, Topology , Curves, Algebraic
Journal or Publication Title: International Journal of Number Theory
Publisher: World Scientific Publishing Co. Pte. Ltd.
ISSN: 1793-0421
Official Date: May 2021
Dates:
DateEvent
May 2021Published
12 August 2020Available
17 July 2020Accepted
Volume: 17
Number: 4
Page Range: pp. 903-923
DOI: 10.1142/S1793042121500147
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): Electronic version of an article published as The proportion of genus one curves over ℚ defined by a binary quartic that everywhere locally have a point. Manjul Bhargava, John Cremona, and Tom Fisher International Journal of Number Theory. [Journal, Volume, Issue, Year, Pages] https://doi.org/10.1142/S1793042121500147 © World Scientific Publishing Company. https://www.worldscientific.com/worldscinet/ijnt
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 5 August 2020
Date of first compliant Open Access: 12 August 2021
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
Investigator Grant Simons Foundationhttp://dx.doi.org/10.13039/100000893
DMS-1001828National Science Foundationhttp://dx.doi.org/10.13039/501100008982
EP/K034383/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
676541Horizon 2020 Framework Programmehttp://dx.doi.org/10.13039/100010661
UNSPECIFIEDStiftung zur Förderung der Reinhold-Würth-Hochschule der Hochschule Heilbronnhttp://dx.doi.org/10.13039/501100005777
Is Part Of: 1
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