# Journal of the Ramanujan Mathematical Society

Volume 33, Issue 2, June 2018 pp. 177–204.

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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