Rank frequencies for quadratic twists of elliptic curves

2000/10/05 by Karl Rubin, Alice Silverberg, Rubin, Karl +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.math/0010056

14 pages

openalex publication_date 2000/10/05 · arxiv created 2001/06/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give explicit examples of infinite families of elliptic curves E over Q with (nonconstant) quadratic twists over Q(t) of rank at least 2 and 3. We recover some results announced by Mestre, as well as some additional families. Suppose D is a squarefree integer and let r_E(D) denote the rank of the quadratic twist of E by D. We apply results of Stewart and Top to our examples to obtain results of the form #D : |D| < x, r_E(D) >= 2 >> x^1/3, #D : |D| < x, r_E(D) >= 3 >> x^1/6 for all sufficiently large x.

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