Class number divisibility for imaginary quadratic fields

TL;DR

This paper refines results on the distribution of class groups in imaginary quadratic fields, providing lower bounds for fields with specific class group elements and applications to elliptic curve twists.

Contribution

It improves Soundararajan's theorem on class group elements in imaginary quadratic fields and connects these results to bounds on elliptic curve twists with large Selmer groups.

Findings

Lower bounds for the number of quadratic fields with class groups containing elements of order g

Extension of results to quadratic twists of elliptic curves with high Selmer rank

Refinement of previous asymptotic estimates for class group distributions

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 $D \equiv A \pmod{B}$ 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 $p \in \{3,5,7\}$.