The first level of $\mathbb{Z}_p$-extensions and compatibility of heuristics

TL;DR

This paper investigates the structure of class groups in the first level of cyclotomic $bZ_p$-extensions over imaginary quadratic fields, linking Cohen–Lenstra–Martinet heuristics with Ellenberg–Jain–Venkatesh predictions.

Contribution

It describes possible class group structures under certain conditions and establishes compatibility between two prominent heuristics in number theory.

Findings

Class group structures are characterized when the $p$-part is cyclic.

Compatibility shown between Cohen–Lenstra–Martinet and Ellenberg–Jain–Venkatesh heuristics.

Results support the predicted frequency of the Iwasawa invariant $\lambda=1$.

Abstract

Let $K$ be an imaginary quadratic field in which the odd prime $p$ does not split. When the $p$-part of the class group of $K$ is cyclic, we describe the possible structures for the $p$-part of the class group of the first level of the cyclotomic $\mathbb{Z}_p$-extension of $K$. This allows us to show the compatibility of the heuristics of Cohen--Lenstra--Martinet for class groups with the heuristics of Ellenberg--Jain--Venkatesh for how often the cyclotomic Iwasawa invariant $\lambda$ equals 1.