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

TL;DR

This paper studies when smooth plane cubic curves over various fields can be represented by determinants of matrices with linear forms, proving positive proportions over certain fields and relating to local-global principles over Q.

Contribution

It proves that all smooth plane cubics over large or ample fields admit linear determinantal representations and establishes positive proportions over Q, linking to conjectures on Selmer groups.

Findings

Any smooth plane cubic over a large or ample field admits a linear determinantal representation.

A positive proportion of smooth plane cubics over Q admit such representations when ordered by height.

Under a conjecture, a positive proportion over Q fail the local-global principle for these 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.