Affiliation:
1. School of Mathematics, Southeast University, Nanjing 210096, China
2. Nanjing University Business School, Nanjing 210093, China
Abstract
Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Algebra and Number Theory
Cited by
2 articles.
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1. On ranks of quadratic twists of a Mordell curve;The Ramanujan Journal;2022-05-13
2. On the rational solutions of y^2 =x^3 + k^{6n+3};Notes on Number Theory and Discrete Mathematics;2021-09