An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields

Abstract

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.