Effect of monogenicity on $2$-torsion in the class group of number fields of odd degree

TL;DR

This paper investigates the average 2-torsion in the class groups of monogenised number fields of odd degree, extending previous cubic field results and showing the doubling phenomenon persists across all odd degrees under certain conditions.

Contribution

It generalizes the doubling phenomenon of 2-torsion in class groups from cubic fields to all odd degrees, providing bounds and conditional results for monogenised fields.

Findings

Average 2-torsion is bounded by twice the Cohen-Lenstra-Martinet-Malle prediction.

Doubling phenomenon persists across all odd degrees under a tail estimate.

Results extend previous cubic field findings to higher odd degrees.

Abstract

We study average $2$-torsion in the class group of monogenised fields of odd degree. Bhargava-Hanke-Shankar have recently shown that the average number of non-trivial $2$-torsion elements in the class group of monogenised cubic fields of a fixed signature is twice the value predicted by the Cohen-Lenstra-Martinet-Malle heuristic for the full family. Fix any odd degree $n$ and signature $(r_{1},r_{2})$. In this work, we prove that the average number of non-trivial elements in the class group of monogenised fields of degree $n$ and signature $(r_{1},r_{2})$ is bounded by twice the value predicted by Cohen-Lenstra-Martinet-Malle. Conditional on a tail estimate for $n \ge 5$, this shows that the doubling phenomenon uncovered for cubic fields by Bhargava-Hanke-Shankar persists across all odd degrees and signatures.