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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
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WRAP-Proportion-genus-curves-defined-binary-quartic-locally-point-Cremona-2020.pdf - Accepted Version - Requires a PDF viewer. Download (802Kb) | Preview |
Official URL: https://doi.org/10.1142/S1793042121500147
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 | ||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||||||||||||
| Divisions: | Faculty of 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: |
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| Volume: | 17 | ||||||||||||||||||
| Number: | 4 | ||||||||||||||||||
| Page Range: | pp. 903-923 | ||||||||||||||||||
| DOI: | 10.1142/S1793042121500147 | ||||||||||||||||||
| Status: | Peer Reviewed | ||||||||||||||||||
| Publication Status: | Published | ||||||||||||||||||
| Publisher Statement: | 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 | ||||||||||||||||||
| RIOXX Funder/Project Grant: |
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