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DC Field | Value | Language |
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dc.contributor.author | Yasuhiro Ishitsuka | - |
dc.date.accessioned | 2024-02-27T05:56:59Z | - |
dc.date.available | 2024-02-27T05:56:59Z | - |
dc.date.issued | 2017 | - |
dc.identifier.uri | http://gnanaganga.inflibnet.ac.in:8080/jspui/handle/123456789/6338 | - |
dc.description.abstract | Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear determinantal representations of smooth plane cubics over various fields, and prove that any smooth plane cubic over a large field (or an ample field) admits a linear determinantal representation. Since local fields are large, any smooth plane cubic over a local field always admits a linear determinantal representation. As an application, we prove that a positive proportion of smooth plane cubics over Q, ordered by height, admit linear determinantal representations. We also prove that, if the conjecture of Bhargava- Kane- Lenstra- Poonen- Rains on the distribution of Selmer groups is true, a positive proportion of smooth plane cubics over Q fail the local-global principle for the existence of linear determinantal representations. | - |
dc.publisher | Journal of the Ramanujan Mathematical Society | - |
dc.title | A Positive Proportion of Cubic Curves Over Q Admit Linear Determinantal Representations | - |
dc.vol | Vol 32 | - |
dc.issued | No 3 | - |
Appears in Collections: | Articles to be qced |
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A positive proportion of cubic curves over Q admit.pdf Restricted Access | 957.3 kB | Adobe PDF | View/Open Request a copy |
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