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

Yasuhiro Ishitsuka, Tetsushi Ito

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.

Original languageEnglish
Pages (from-to)141-162
Number of pages22
JournalRamanujan Journal
Volume43
Issue number1
DOIs
Publication statusPublished - May 1 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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