2015/12/16 by Yasuhiro Ishitsuka, Ishitsuka, Yasuhiro · 1 citation
Mathematics · #11D41 #14F22 #14H50 (Primary) #14K15 #14K30 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1512.05167
openalex publication_date 2015/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Can a smooth plane cubic be defined by the determinant of a square matrix\nwith entries in linear forms in three variables? If we can, we say that it\nadmits a linear determinantal representation. In this paper, we investigate\nlinear determinantal representations of smooth plane cubics over various\nfields, and prove that any smooth plane cubic over a large field (or an ample\nfield) admits a linear determinantal representation. Since local fields are\nlarge, any smooth plane cubic over a local field always admits a linear\ndeterminantal representation. As an application, we prove that a positive\nproportion of smooth plane cubics over Q, ordered by height, admit linear\ndeterminantal representations. We also prove that, if the conjecture of\nBhargava-Kane-Lenstra-Poonen-Rains on the distribution of Selmer groups is\ntrue, a positive proportion of smooth plane cubics over Q fail the local-global\nprinciple for the existence of linear determinantal representations.\n