Linear determinantal representations of smooth plane cubics over finite\n fields

2016/03/31 by Yasuhiro Ishitsuka, Ishitsuka, Yasuhiro
Computer Science · Mathematics · #11D25 #12Y05 #14G15 #14H50 (Primary) #15A33 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1604.00115

openalex publication_date 2016/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Orbit parametrizations of theta characteristics on hypersurfaces over\n arbitrary fields
• A positive proportion of cubic curves over Q admit linear determinantal\n representations

Related

• An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields
• A positive proportion of cubic curves over Q admit linear determinantal\n representations
• On algorithms to obtain linear determinantal representations of smooth\n plane curves of higher degree
• Determinantal representations of smooth cubic surfaces
• Determinantal representations of the cubic discriminant