Rank Frequencies for Quadratic Twists of Elliptic Curves

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 $x$.