On Malle's conjecture for nilpotent groups, I

TL;DR

This paper develops a framework to study Malle's conjecture for nilpotent groups, proving it for certain classes and providing bounds, supported by heuristic arguments, advancing understanding of number field distributions.

Contribution

Introduces an abstract framework for Malle's conjecture in nilpotent groups and proves the conjecture for groups with central elements of order p, also providing bounds and heuristics.

Findings

Proved the strong form of Malle's conjecture for nilpotent groups with all elements of order p central.

Provided an upper bound for nilpotent groups tight up to logarithmic factors.

Developed a heuristic argument supporting Malle's conjecture for nilpotent groups.

Abstract

We develop an abstract framework for studying the strong form of Malle's conjecture for nilpotent groups $G$ in their regular representation. This framework is then used to prove the strong form of Malle's conjecture for any nilpotent group $G$ such that all elements of order $p$ are central, where $p$ is the smallest prime divisor of $\# G$. We also give an upper bound for any nilpotent group $G$ tight up to logarithmic factors, and tight up to a constant factor in case all elements of order $p$ pairwise commute. Finally, we give a new heuristical argument supporting Malle's conjecture in the case of nilpotent groups in their regular representation.