On Malle’s conjecture for nilpotent groups

Koymans, Peter; Pagano, Carlo

doi:10.1090/btran/140

On Malle’s conjecture for nilpotent groups
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by Peter Koymans and Carlo Pagano

Trans. Amer. Math. Soc. Ser. B 10 (2023), 310-354

DOI: https://doi.org/10.1090/btran/140

Published electronically: March 3, 2023

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Abstract:

We develop an abstract framework for studying the strong form of Malle’s conjecture [J. Number Theory 92 (2002), pp. 315–329; Experiment. Math. 13 (2004), pp. 129–135] 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.

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