An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields

TL;DR

This paper presents an algorithm for computing all linear determinantal representations of smooth plane cubics over finite fields, connecting classical interpretations with modern computational methods.

Contribution

It introduces a new algorithm to find all linear determinantal representations of smooth plane cubics, including over finite fields and rational numbers.

Findings

Algorithm successfully computes all representations over finite fields.

Provides a method to classify representations up to equivalence.

Includes recent computations for twisted Fermat cubics over rationals.

Abstract

We give a brief report on our computations of linear determinantal representations of smooth plane cubics over finite fields. After recalling a classical interpretation of linear determinantal representations as rational points on the affine part of Jacobian varieties, we give an algorithm to obtain all linear determinantal representations up to equivalence. We also report our recent study on computations of linear determinantal representations of twisted Fermat cubics defined over the field of rational numbers. This paper is a summary of the author's talk at the JSIAM JANT workshop on algorithmic number theory in March, 2016. Details will appear elsewhere.