Rank frequencies for quadratic twists of elliptic curves

Details Rank frequencies for quadratic twists of elliptic curves Rank frequencies for quadratic twists of elliptic curves

2000-10-05

arxiv.org/abs/math/0010056v3

Mathematics

Number Theory

14 pages

Scientific paper

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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