Journal of the Ramanujan Mathematical Society
Volume 32, Issue 3, September 2017 pp. 231–257.
A positive proportion of cubic curves over Q admit
linear determinantal representations
Authors:
Yasuhiro Ishitsuka
Author institution:Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502,
Japan
Summary:
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.
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