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 $4\nmid g$ or if $A$ and $B$ satisfy certain conditions, then the number of negative square-free $D\equiv A(\mathrm{mod}B)$ 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 $p\in \{3,5,7\}$.

11 pages