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.

arXiv comment28 pages

References in corpus (5)

• Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface
• Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples
• Automorphisms of elliptic $K3$ surfaces and Salem numbers of maximal degree
• Elliptic K3 surfaces with geometric Mordell-Weil rank 15
• Automorphisms of supersingular K3 surfaces and Salem polynomials

Cited by in corpus (2)

• Common valuations of division polynomials
• Mordell-Weil ranks of families of elliptic curves parametrized by binary quadratic forms