On rationality of the intersection points of a line with a plane quartic

2010/06/04 by Roger Oyono, Christophe Ritzenthaler, Oyono, Roger +1
Computer Science · Mathematics · #11G20 #14G05 #14G15 #14H45 #14N10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1006.0873

openalex publication_date 2010/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the rationality of the intersection points of certain lines and smooth plane quartics C defined over F_q. For q ≥ 127, we prove the existence of a line such that the intersection points with C are all rational. Using another approach, we further prove the existence of a tangent line with the same property as soon as the characteristic of F_q is different from 2 and q ≥ 66²+1. Finally, we study the probability of the existence of a rational flex on C and exhibit a curious behavior when the characteristic of F_q is equal to 3.

Citations

• Elliptic curve cryptosystems

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