Quadratic twists of genus one curves

2024/01/17 by Lukáš Novák, Novak, Lukas · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.09626

openalex publication_date 2024/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given irreducible and monic polynomial f(x) ∈ ℤ[x] of degree 4, we consider the quadratic twists by square-free integers q of the genus one quartic H : y²=f(x) H_q : qy²=f(x). We say that a curve C is everywhere locally soluble (ELS) if it has a solution in ℝ and in ℚ_p for every prime p (i.e. if C(ℝ)≠ ∅ and C(ℚ_p)≠ ∅ for all primes p). Let L=\q∈ ℕ : q is square-free and H_q is ELS\ denote the set of positive square-free integers q for which H_q is everywhere locally soluble. For a real number x let L(x)= #\q∈ L: q

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