Invariants for A4 fields and the Cohen-Lenstra heuristics

TL;DR

This paper investigates deviations from Cohen-Lenstra heuristics in A4 fields with roots of unity, proposing an invariant in the Schur multiplier group to explain observed discrepancies and providing computational evidence for specific cases.

Contribution

It introduces an invariant in the Schur multiplier group to explain deviations from Cohen-Lenstra heuristics in A4 fields with roots of unity, and simplifies its computation in certain cases.

Findings

The invariant explains the discrepancy in the number of cyclic cubic fields with specific 2-class groups.

Simplified formulas for the invariant are provided for ramified cubic fields.

Computational results up to discriminant 10^8 support the proposed explanation.

Abstract

This article discusses deviations from the Cohen-Lenstra heuristics when roots of unity are present. In particular, we propose an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is $C_2\times C_2$ and the number predicted by the Cohen-Lenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute the invariant when the cubic field is ramified at exactly one prime, up to $10^8$.