Specializations of a generic rank-2 curve of Shioda

Abstract

In a paper from twenty years ago, Brown and Myers proved the clean and uniform result that if m>1 is an integer, then every curve of the form y2=x3−x+m2 has Mordell-Weil rank at least 2. In this article, we further our investigation into the more general family of curves, in which the polynomial x3−x is replaced by an arbitrary cubic which is squarefree and has three integral roots. In particular, a new strategy will unveil a subfamily of curves with surprisingly large average rank, and one startling example.