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Title: | A Positive Proportion of Cubic Curves Over Q Admit Linear Determinantal Representations |
Authors: | Yasuhiro Ishitsuka |
Issue Date: | 2017 |
Publisher: | Journal of the Ramanujan Mathematical Society |
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. |
URI: | http://gnanaganga.inflibnet.ac.in:8080/jspui/handle/123456789/6338 |
Appears in Collections: | Articles to be qced |
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