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

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.

The local–global principle for symmetric determinantal representations of smooth plane curves in characteristic two

“…In that case, there is no obstruction accounted by the Brauer group Br(K) (or the relative Brauer group Br(C/K)). In [14], [15], the first author gave an algorithm to calculate linear (i.e. not necessarily symmetric) determinantal representations of smooth plane cubics with rational points.…”

“…Details on the calculation of symmetric and linear determinantal representations of twisted Hesse cubics will appear elsewhere. Some examples over Q are given in [15,Section 6] for m = 0 (i.e. the case of "twisted Fermat cubics").…”