1. Introduction. By Faltings' theorem, a (smooth complete geometrically irreducible) curve of genus > 1 over a number field has finitely many rational points. By [2], it is widely believed that the number of these rational points is bounded in terms of the genus. In By Silverman's result, given a curve of genus > 1 over a number field, finding infinitely many twists with a bounded number of rational points becomes a problem of finding infinitely many twists with bounded MordellWeil rank. Even for special cases such as Thue equations (see [11]), an answer to this problem is sometimes not known. For the case of elliptic curves, by Kolyvagin's result [7] and the modularity of elliptic curves proved by Wiles et al., results such as [14], in which quadratic twists with analytic rank 0 are computed, imply that given an elliptic curve over Q, there are infinitely many quadratic twists with Mordell-Weil rank 0, i.e., algebraic rank 0. There are also results of this type such as , [24], [25], and [3] which rather directly show that there is a "positive proportion" of algebraic rank-0 quadratic twists of certain elliptic curves.In this paper, we consider a family of twists of superelliptic curves over a global field, and obtain results about the distribution of a certain Selmer rank in this family of twists. These results imply that for these twists, the problem of finding infinitely many twists with bounded Mordell-Weil rank has a positive answer and, hence, there are infinitely many twists with bounded number of rational points if the genus is > 1. Our result can be applied to Thue equations which can be mapped down to superelliptic curves considered in this paper. For the case of superelliptic curves over a constant

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Let E be the elliptic curve given by a Mordell equationStoll found a precise formula for the size of a Selmer group of E for certain values of A. For D ∈ Z, let E D denote the quadratic twist Dy 2 = x 3 − A. We use Stoll's formula to show that for a positive square-free integer A ≡ 1 or 25 mod 36 and for a nonnegative integer k, we can compute a lower bound for the proportion of square-free integers D up to X such that rank E D (Q) 2k. We also compute an upper bound for a certain average rank of quadratic twists of E.

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Let E/Q be an elliptic curve with no nontrivial rational 2-torsion points where Q is the rational numbers. In this paper, we prove that there is a quadratic twist E D for which the rank of the 2-Selmer group is less than or equal to 1. By the author's earlier result [1], this implies an unconditional distribution result on the size of the 2-part of the Tate-Shafarevich group of the quadratic twists of E. By [7], if we assume the finiteness of the Tate-Shafarevich group of an elliptic curve, our result yields a fairly general distribution result for quadratic twists with Mordell-Weil rank 1. IntroductionLet E/Q be an elliptic curve given by y 2 = x 3 + ax + b, E D , a quadratic twist D y 2 = x 3 + ax + b, and Sel (2) (E D ), the 2-Selmer group of E D . In this paper, we prove Theorem 1.1 1 Let E/Q be an elliptic curve with no nontrivial rational 2-torsion points. Then,This result will follow from [1, Theorem 1.2] once we prove the existence of D such that dim Sel (2) (E D ) ≤ 1. Let III(E D )[2] and rank E D (Q) denote the 2-part of the Tate-Shafarevich group and the Mordell-Weil rank, respectively. Sinceour theorem implies the obvious distribution results for #III(E D )[2] and rank E D (Q). The distribution result for rank E D (Q) = 0 is also obtained in [9] and [8] by establishing the non-vanishing of L-functions, 1 The result is improved recently by Mazur and Rubin [6] using Kramer's work [5]. arXiv:0809.5019v3 [math.NT] 10 Jun 2009 2 Computing the 2-Selmer groupRecall that E/Q is given by y 2 = x 3 + ax + b, and does not have nontrivial rational 2-torsion points. Let z 1 , z 2 , and z 3 be the x-coordinates of the 2-torsion points in Q, i.e., the roots of x 3 + ax + b. Let L = L E be the field extension Q(z 1 ). For each place p, let us denote by n p the number of places of O L lying over p. Let S E be the set of places of Q consisting of ∞ and 2, and places of bad reduction of E/Q. By [10, Proposition 3.4], we have the following isomorphism: