Distribution of Mordell–Weil ranks of families of elliptic curves
Volume 108 / 2016
Banach Center Publications 108 (2016), 201-229
MSC: Primary 14H52; Secondary 11D25, 11D45, 11G05.
DOI: 10.4064/bc108-0-16
Abstract
We discuss the distribution of Mordell–Weil ranks of the family of elliptic curves $y^2=(x+\alpha f^2)(x+\beta b g^2)(x+\gamma h^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and $\alpha,\beta,\gamma$ are elements from $\overline{\mathbb{Q}}$. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
