A note on the distribution of Iwasawa invariants of imaginary quadratic fields

TL;DR

This paper explores the relationship between class group ranks and Iwasawa invariants in imaginary quadratic fields, providing statistical insights into their distribution, with some results relying on heuristics and others being unconditional.

Contribution

It establishes a new connection between class group $p$-ranks and Iwasawa $ ext{lambda}$-invariants, and derives statistical distribution results for these invariants in imaginary quadratic fields.

Findings

Distribution of $ ext{lambda}$-invariants analyzed statistically.

Conditional results based on Cohen--Lenstra heuristics.

Unconditional results derived from existing theorems.

Abstract

Given an odd prime number $p$ and an imaginary quadratic field $K$, we establish a relationship between the $p$-rank of the class group of $K$, and the classical $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $K$. Exploiting this relationship, we prove statistical results for the distribution of $\lambda$-invariants for imaginary quadratic fields ordered according to their discriminant. Some of our results are conditional since they rely on the original Cohen--Lenstra heuristics for the distribution of the $p$-parts of class groups of imaginary quadratic fields. Some results are unconditional results ad are obtained by leveraging theorems of Byeon, Craig and others.