Abstract
We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.
Original language | English |
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Article number | 71 |
Journal | Seminaire Lotharingien de Combinatoire |
Issue number | 84 |
State | Published - 2020 |
Bibliographical note
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Funding
∗[email protected]. Partially supported by the Irish Research Council (GOIPD/2018/319). †[email protected]. ‡[email protected]. Partially supported by The Emmy Noether Minerva Research Institute at Bar-Ilan University during the preliminary stages of this project The research of all three authors was supported by a grant from the GIF, the German– Israeli Foundation for Scientific Research and Development (1246/2014)
Funders | Funder number |
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Emmy Noether Minerva Research Institute at Bar-Ilan University | |
German-Israeli Foundation for Scientific Research and Development | 1246/2014 |
Irish Research Council | GOIPD/2018/319 |
Keywords
- Igusa functions
- combinatorial reciprocity theorem
- normal zeta functions
- subgroup growth
- weak orders