Indivisibility of class numbers of imaginary quadratic fields

TL;DR

This paper provides an estimate for the count of imaginary quadratic fields with class numbers indivisible by a prime, matching Cohen-Lenstra heuristics, and applies it to elliptic curve rank 0 twists.

Contribution

It generalizes Wiles' theorem by quantifying the distribution of class numbers indivisible by a prime under local conditions, aligning with Cohen-Lenstra heuristics.

Findings

Estimate matches Cohen-Lenstra heuristics

Quantifies class number indivisibility for negative discriminants

Applies results to elliptic curve rank 0 twists

Abstract

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen-Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.