Normal zeta functions of the Heisenberg groups over number rings. I: The unramified case. (English) Zbl 1315.20023
The aim of this paper is to compute the local factors of normal zeta functions of the Heisenberg group at unramified rational primes, and establish functional equations for these local zeta functions upon the inversion of the prime. For a general setting, let \(G\) be a finitely generated group and \(a_n(G)\) be the number of normal subgroups of index \(n\) in \(G\), which is finite. Then \(\zeta_G(s)\), the normal zeta function of \(G\), is the Dirichlet generating function for the sequence \(a_n(G)\), \(n\in\mathbb N\). If moreover \(G\) is nilpotent then \(\zeta_G(s)\) has an Euler product decomposition where the local factor \(\zeta_{G,p}(s)\) counts normal subgroups of \(p\)-power index in \(G\), for each prime \(p\).
The authors consider the particular case where \(G\) is the two-step Heisenberg group \(H(\mathcal O_K)\) over the ring of integers of a number field \(K\). Assume that a prime \(p\) is of decomposition type \((\mathbf{e,f})\) where the vectors \(\mathbf{e,f}\) resemble the ramification indices and the inertia degrees, respectively. In [F. J. Grunewald et al., Invent. Math. 93, No. 1, 185-223 (1988; Zbl 0651.20040)] it is proved that given \(\mathbf{e,f}\) with \(\mathbf e\cdot\mathbf f=n\), there exists a rational function \(W_{\mathbf{e,f}}(X,Y)\) such that \(\zeta_{H(\mathcal O_K),p}(s)=W_{\mathbf{e,f}}(p,p^{-s})\) for all number fields \(K\) of degree \(n\) and all primes \(p\) of decomposition type \((\mathbf{e,f})\) in \(K\).
The authors make a general conjecture on a functional equation for \(W_{\mathbf{e,f}}(X,Y)\). In the paper under review they compute \(W_{\mathbf{e,f}}(X,Y)\) and prove this conjecture in the case that \(p\) is unramified, i.e., all the \(e_i\)’s are equal to 1. In a subsequent paper they also prove the conjecture in the case that \(p\) is non-split. These results give strong supporting evidence for the general conjecture.
The proof is highly combinatorial and technical. Roughly speaking, in the unramified case, the authors manage to express the local zeta functions as sums, indexed by Dyck words, of functions defined in terms of certain combinatorial objects called weak orderings. They also mention a few related open problems such as computing \(W_{\mathbf{e,f}}(X, Y)\) for general \(\mathbf e\), and more generally computing the normal zeta functions of other finitely generated nilpotent groups. Another interesting direction is to count all subgroups of finite index, not only normal subgroups. The whole project is much bigger and there are lots of things to be done.
The authors consider the particular case where \(G\) is the two-step Heisenberg group \(H(\mathcal O_K)\) over the ring of integers of a number field \(K\). Assume that a prime \(p\) is of decomposition type \((\mathbf{e,f})\) where the vectors \(\mathbf{e,f}\) resemble the ramification indices and the inertia degrees, respectively. In [F. J. Grunewald et al., Invent. Math. 93, No. 1, 185-223 (1988; Zbl 0651.20040)] it is proved that given \(\mathbf{e,f}\) with \(\mathbf e\cdot\mathbf f=n\), there exists a rational function \(W_{\mathbf{e,f}}(X,Y)\) such that \(\zeta_{H(\mathcal O_K),p}(s)=W_{\mathbf{e,f}}(p,p^{-s})\) for all number fields \(K\) of degree \(n\) and all primes \(p\) of decomposition type \((\mathbf{e,f})\) in \(K\).
The authors make a general conjecture on a functional equation for \(W_{\mathbf{e,f}}(X,Y)\). In the paper under review they compute \(W_{\mathbf{e,f}}(X,Y)\) and prove this conjecture in the case that \(p\) is unramified, i.e., all the \(e_i\)’s are equal to 1. In a subsequent paper they also prove the conjecture in the case that \(p\) is non-split. These results give strong supporting evidence for the general conjecture.
The proof is highly combinatorial and technical. Roughly speaking, in the unramified case, the authors manage to express the local zeta functions as sums, indexed by Dyck words, of functions defined in terms of certain combinatorial objects called weak orderings. They also mention a few related open problems such as computing \(W_{\mathbf{e,f}}(X, Y)\) for general \(\mathbf e\), and more generally computing the normal zeta functions of other finitely generated nilpotent groups. Another interesting direction is to count all subgroups of finite index, not only normal subgroups. The whole project is much bigger and there are lots of things to be done.
Reviewer: Dongwen Liu (Zhejiang)
MSC:
| 20E07 | Subgroup theorems; subgroup growth |
| 11M41 | Other Dirichlet series and zeta functions |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A15 | Exact enumeration problems, generating functions |
| 20G30 | Linear algebraic groups over global fields and their integers |
| 20F18 | Nilpotent groups |
Keywords:
normal zeta functions; subgroups of finite index; Heisenberg groups; functional equations; rings of integers of number fieldsCitations:
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