On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves

Abstract

Let $l$ be the prime $3,5$ or $7$ and let $m$ be a nonzero integer. We give a method for constructing an infinite family of pairs of quadratic fields ${\mathbb Q} \bigl(\sqrt D \big)$ and ${\mathbb Q} \bigl(\sqrt{mD} \big)$ with both class numbers divisible by $l$. Such quadratic fields are parametrized by rational points on a specified elliptic curve.