On the arithmetic of twists of superelliptic curves

Abstract

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

Ranks of twists of elliptic curves and Hilbert’s tenth problem

“…When K = Q, a version of Theorem 1.4 was proved by Chang in [Ch1,Theorem 4.10]. Also in the case K = Q, Chang has proved (slightly weaker) versions of Theorem 1.7 and Corollary 1.12 below, namely [Ch2,Theorem 1.1] and [Ch2,Corollary 1.2], respectively.…”

Note on the rank of quadratic twists of Mordell equations

“…This observation is generalized for superelliptic curves over global fields in [1]. To my knowledge, the only known example of an elliptic curve with infinitely many cubic twists of Mordell-Weil rank 0 is x 3 + y 3 = D proved by D. Lieman [6].…”

Quadratic twists of elliptic curves with small Selmer rank

“…These isomorphisms are defined with a choice of representatives of Gal(Q/Q)orbits or Gal(Q p /Q p )-orbits in E [2]. With certain choices of representatives, restriction maps res p : H 1 (Q, E [2]) → H 1 (Q p , E [2]) extend to the natural maps L * /(L * ) 2 → L * p /(L * p ) 2 which we also denote by res p (see [1,Proposition 2.4]).…”

“…. Let us identify the cohomology group H 1 (Q, E [2]) S E with the kernel in (1). Let q 0 = ∞, and q 1 = 2, and write S E := {q 0 , q 1 , .…”

“…Now let D = pD 0 . Then, at each r contained in S E , the local coboundary image of E D is equal to that of E at r (see [1,Proposition 2.5]). Let T E D be T E ∪ {p}.…”