Quadratic twists of genus one curves and Diophantine quintuples

2022/09/26 by Matija Kazalicki, Kazalicki, Matija · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2209.12864

openalex publication_date 2022/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the theory of Diophantine m-tuples, we study rational points on quadratic twists H^d:d y²=(x²+6x-18)(-x²+2x+2), where |d| is a prime. If we denote by S(X)=\ d ∈ ℤ: H^d(ℚ)≠ ∅, |d| \textrm is a prime\textrm and |d| < X\, then, by assuming some standard conjectures about the ranks of elliptic curves in the family of quadratic twists, we prove that as X → ∞ (43)/(256)+o(1)≤ (#S(X))/(2π(X))≤ (46)/(256)+o(1).

Cited by

• Twists arising from torsion points

Related

• Ranks of Elliptic Curves Twisted by Quadratic Forms
• Divisibility by 2 on quartic models of elliptic curves and rational Diophantine D(q)-quintuples
• Quadratic twists of pairs of elliptic curves
• Quadratic twists of genus one curves
• Quadratic Twists of Elliptic Curves