Linear determinantal representations of smooth plane cubics over finite fields

TL;DR

This paper investigates linear determinantal representations of smooth plane cubics over finite fields, providing explicit formulas, counting methods, and classifications based on the number of such representations.

Contribution

It offers an explicit formula for representations, counts classes using Schoof's formula, and classifies cubics by their determinantal representation counts.

Findings

Explicit formulas for representations

Counting of classes using Schoof's formula

Classification of cubics with 0, 1, or 2 representations

Abstract

In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count the number of projective equivalence classes of smooth plane cubics over a finite field admitting prescribed number of equivalence classes of linear determinantal representations. As an application, we determine isomorphism classes of smooth plane cubics over a finite field with 0, 1 or 2 equivalence classes of linear determinantal representations.