A positive proportion of cubic curves over Q admit linear determinantal representations

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.