Class numbers in cyclotomic Z_p-extensions

TL;DR

This paper investigates the triviality of ideal class groups in layers of cyclotomic Z_p-extensions over the rationals, providing evidence supporting a conjecture that these groups are trivial for all primes p and positive integers n.

Contribution

The paper offers new evidence and heuristics supporting the conjecture that class groups in all layers of cyclotomic Z_p-extensions are trivial, extending previous results and insights.

Findings

Evidence supporting the conjecture of trivial class groups in all layers

Refined heuristics based on Cohen-Lenstra predictions

New results on class numbers of specific B(p,n) layers

Abstract

For any prime p and any positive integer n, let B(p,n) denote the n-th layer of the cyclotomic Z_p-extension over the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular B(p,n), we provide evidence for the following conjecture: For all primes p and positive integers n, the ideal class group of B(p,n) is trivial.