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2001 Rank Frequencies for Quadratic Twists of Elliptic Curves
Karl Rubin, Alice Silverberg
Experiment. Math. 10(4): 559-570 (2001).

Abstract

We give explicit examples of infinite families of elliptic curves $E$ over $\funnyQ$ with (nonconstant) quadratic twists over $\funnyQ(t)$ 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) \ge 2\} \gg x^{1/3},

{\#\{D : |D|<x, \, r_E(D) \ge 3\} \gg x^{1/6}}

for all sufficiently large $x$.

Citation

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Karl Rubin. Alice Silverberg. "Rank Frequencies for Quadratic Twists of Elliptic Curves." Experiment. Math. 10 (4) 559 - 570, 2001.

Information

Published: 2001
First available in Project Euclid: 26 November 2003

zbMATH: 1035.11025
MathSciNet: MR1881757

Rights: Copyright © 2001 A K Peters, Ltd.

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Vol.10 • No. 4 • 2001
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