Class number divisibility for imaginary quadratic fields

Abstract

In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let $$A,B,g \ge 3$$ be positive integers such that $$\gcd (A,B)$$ is square-free. We refine Soundararajan’s result to show that if $$4 \not \mid g$$ or if A and B satisfy certain conditions, then the number of negative square-free $$D \equiv A \pmod {B}$$ down to $$-X$$ such that the ideal class group of $${\mathbb {Q}} (\sqrt{D})$$ contains an element of order g is bounded below by $$X^{\frac{1}{2} + \epsilon (g) - \epsilon }$$, where the exponent is the same as in Soundararajan’s theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with p-Selmer group of rank at least 2, where $$p \in \{3,5,7\}$$.