On Malle’s conjecture for nilpotent groups

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