Distribution of Mordell--Weil ranks of families of elliptic curves

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.

28 pages

References in corpus (3)

• Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface
• Automorphisms of elliptic $K3$ surfaces and Salem numbers of maximal degree
• Elliptic K3 surfaces with geometric Mordell-Weil rank 15

Cited by in corpus (1)

• Mordell-Weil ranks of families of elliptic curves parametrized by binary quadratic forms