Pro-isomorphic zeta functions of some $D^{\ast}$ Lie lattices of even rank

Let $G$ be a finitely generated group. The pro-isomorphic zeta function of $G$, which was originally introduced by Grunewald, Segal, and Smith [14], is the Dirichlet series $\zeta^{\wedge}_{G}(s)=\sum_{m=0}^{\infty}a_{m}^{\wedge}(G)m^{-s}$. Here $s$ is a complex variable and $a_{m}^{\wedge}(G)$ is the (necessarily finite) number of subgroups $H\leq G$ of index $m$ such that the profinite completion of $H$ is isomorphic to that of $G$. In practice it is convenient to interpret this series as counting linear objects. Let $\mathcal{L}$ be a $\mathbb{Z}$-algebra, which for our purposes is a free $\mathbb{Z}$-module of finite rank endowed with a $\mathbb{Z}$-bilinear multiplication. Its pro-isomorphic zeta function is the Dirichlet series $\zeta^{\wedge}_{\mathcal{L}}(s)=\sum_{m=0}^{\infty}b^{\wedge}_{m}(\mathcal{L})m^{-s}$, where $b^{\wedge}_{m}(\mathcal{L})$ is the number of subalgebras $\mathcal{M}\leq\mathcal{L}$ of index $n$ such that $\mathcal{M}\otimes\mathbb{Z}_{p}\simeq\mathcal{L}\otimes\mathbb{Z}_{p}$ for all primes $p$. An elementary but fundamental result [14, Proposition 4] is the Euler decomposition $\zeta^{\wedge}_{\mathcal{L}}(s)=\prod_{p}\zeta^{\wedge}_{\mathcal{L},p}(s)$, where $\zeta^{\wedge}_{\mathcal{L},p}(s)$ counts only subalgebras of $p$-power index or, equivalently, $\mathbb{Z}_{p}$-subalgebras of $\mathcal{L}\otimes\mathbb{Z}_{p}$ that are isomorphic to $\mathcal{L}\otimes\mathbb{Z}_{p}$. An analogous decomposition holds for finitely generated torsion-free nilpotent groups. If $G$ is such a group, then there is a Lie lattice $\mathcal{L}(G)$, namely a $\mathbb{Z}$-algebra whose multiplication is a Lie bracket, such that $\zeta^{\wedge}_{\mathcal{L}(G),p}(s)=\zeta^{\wedge}_{G,p}(s)$ for all but finitely many $p$. If $G$ is of class two, then this equality holds for all primes $p$; see, for instance, [14, §4] and [3, §2.1].

The present work computes the pro-isomorphic zeta functions of many members of a certain family of class-two-nilpotent Lie lattices of even rank considered by Berman, Klopsch, and Onn [5]. This family corresponds to the representatives constructed by Grunewald and Segal [13] of commensurability classes of $D^{\ast}$-groups of even Hirsch length; see Section 1.5.1 below.

We now present our main results more precisely. Let $\Delta(x)\in\mathbb{Z}[x]$ be a primary polynomial, i.e. $\Delta(x)=f(x)^{\ell}$ for an irreducible monic polynomial $f(x)$ and $\ell\in\mathbb{N}$. If $\Delta(x)=x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$ for $a_{i}\in\mathbb{Z}$, recall its companion matrix

Let $\mathcal{L}_{\Delta}$ be the Lie lattice of rank $2n+2$ with basis $x_{1},\dots,x_{n},y_{1},\dots,y_{n},z_{1},z_{2}$ and Lie bracket determined by the following:

• •

  $[x_{i},x_{j}]=[y_{i},y_{j}]=0$ for all $1\leq i,j\leq n$;

• •

  $[x_{i},y_{j}]=\delta_{ij}z_{1}+(C_{\Delta})_{ij}z_{2}$ for all $1\leq i,j\leq n$, where $\delta_{ij}$ is the Kronecker delta;

• •

  $z_{1}$ and $z_{2}$ lie in (and indeed span) the center of $\mathcal{L}_{\Delta}$.

We consider the case where $\Delta(x)=f(x)$ is an irreducible polynomial of degree $n\geq 2$ and determine $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$ for all but finitely many $p$. Indeed, let $\beta$ be a root of $f(x)$ and consider the number field $K_{f}=\mathbb{Q}(\beta)$. Recall that the conductor $\mathcal{F}_{f}$ is the largest ideal of the ring of integers $\mathcal{O}_{K_{f}}$ that is contained in $\mathbb{Z}[\beta]$. For all primes $p$ coprime to $\mathcal{F}_{f}$, we compute $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$ explicitly when $n\geq 3$. Theorem 1.4 treats the case $n=2$, which was actually treated 35 years ago, under a different name, by Grunewald, Segal, and Smith.

Moreover, we prove the following finite uniformity statement. Suppose that $K$ is a number field, $p$ is a prime, and $\mathbf{e}=(e_{1},\dots,e_{r})$ and $\mathbf{f}=(f_{1},\dots,f_{r})$ are vectors of natural numbers. We say that $p$ has decomposition type $(\mathbf{e},\mathbf{f})$ in $K$ if $p\mathcal{O}_{K}=\mathfrak{p}_{1}^{e_{1}}\cdots\mathfrak{p}_{r}^{e_{r}}$, where the $\mathfrak{p}_{i}\triangleleft\mathcal{O}_{K}$ are distinct prime ideals with residue fields of cardinality $|\mathcal{O}_{K}/\mathfrak{p}_{i}|=p^{f_{i}}$ for every $1\leq i\leq r$. This implies that $n=\sum_{i=1}^{r}e_{i}f_{i}$. Let $\mathbf{1}$ denote the vector $(1,\dots,1)$.

In fact, in (3.4) below we realize the functions $W_{\mathbf{1},\mathbf{f}}$ as specializations of combinatorially defined functions $\Phi_{r}$ in $2^{r}$ variables introduced for every $r\in\mathbb{N}$ in Definition 2.1. While these $\Phi_{r}$ are reminiscent of some functions that have appeared recently in the literature in the context of enumerative problems arising from algebra [21, 8, 20, 18], they do not seem to be special cases of them. Then (1.1) is immediate from Proposition 2.2, which proves a self-reciprocity of the functions $\Phi_{r}$ under inversion of the variables.

We illustrate the explicit formulas of Theorem 1.1 in a simple example:

Note that the rational function governing $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$ for the ramified primes $p\in\{2,3\}$ does not satisfy a functional equation for any symmetry factor.

For completeness, we state the pro-isomorphic zeta functions $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$ at all but finitely many primes when $f(x)\in\mathbb{Z}[x]$ is an irreducible monic quadratic polynomial. Note that there are only three decomposition types for a prime in a quadratic number field: inert ($(\mathbf{e},\mathbf{f})=((1),(2))$), totally split ($(\mathbf{e},\mathbf{f})=((1,1),(1,1))$) and totally ramified ($(\mathbf{e},\mathbf{f})=((2),(1))$). The following claim is essentially due to Grunewald, Segal, and Smith [14] and is analogous to Theorem 1.1.

Observe that the local pro-isomorphic zeta functions $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$ appearing in Theorem 1.4 decompose as products of factors parametrized by primes of $K_{f}$ dividing $p$. This is a special case of a general phenomenon [3, Proposition 3.14]. The Lie algebras $\mathcal{H}\otimes_{\mathbb{Z}}\mathbb{Q}_{p}$ satisfy a rigidity property [3, Definition 3.8] originally introduced by Segal [22]; as a consequence, the pro-isomorphic zeta function $\zeta^{\wedge}_{\mathcal{H}\otimes\mathcal{O}_{K},p}(s)$ may be computed easily for any number field $K$. Such rigidity does not hold for the Lie algebras $\mathcal{L}_{f}\otimes_{\mathbb{Z}}\mathbb{Q}_{p}$ of Theorem 1.1; this is essentially a consequence of the arithmetic of the number field $K_{f}$, which is larger than $\mathbb{Q}$, controlling the local pro-isomorphic zeta functions $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$.

It is a simple but fundamental observation that computations of local factors of pro-isomorphic zeta functions can be reduced to $p$-adic integrals of a certain form. Consider the $\mathbb{Q}$-Lie algebra $L_{\Delta}=\mathcal{L}_{\Delta}\otimes_{\mathbb{Z}}\mathbb{Q}$, and let $\mathbf{G}_{\Delta}$ be its algebraic automorphism group. This is the algebraic group defined over $\mathbb{Q}$ characterized by the property that $\mathbf{G}_{\Delta}(E)\simeq\mathrm{Aut}_{E}(L_{\Delta}\otimes_{\mathbb{Q}}E)$ for every field $E$ of characteristic zero. Fixing the ordered basis $(x_{1},\dots,x_{n},y_{1},\dots,y_{n},z_{1},z_{2})$ of $\mathcal{L}_{\Delta}$ gives an embedding $\mathbf{G}_{\Delta}\hookrightarrow\mathrm{GL}_{2n+2}$. Now set $G_{\Delta}^{+}(\mathbb{Q}_{p})=\mathbf{G}_{\Delta}(\mathbb{Q}_{p})\cap\mathrm{M}_{2n+2}(\mathbb{Z}_{p})$, and let $G_{\Delta}(\mathbb{Z}_{p})=\mathbf{G}_{\Delta}(\mathbb{Q}_{p})\cap\mathrm{GL}_{2n+2}(\mathbb{Z}_{p})$. Let $\mu$ be the right Haar measure on the group $\mathbf{G}_{\Delta}(\mathbb{Q}_{p})$, normalized so that $\mu(G_{\Delta}(\mathbb{Z}_{p}))=1$. Then by [14, Proposition 3.4] we have

where $|\,\cdot\,|_{p}$ is the normalized valuation on $\mathbb{Q}_{p}$. The structure of $\mathbf{G}_{\Delta}$, for all primary polynomials $\Delta(x)=f(x)^{\ell}$, was determined by Berman, Klopsch, and Onn; see Proposition 3.1 below for the case $\mathrm{deg}\,f(x)\geq 3$. When $\Delta(x)=f(x)$ is irreducible, the domain of integration of (1.2) is sufficiently simple that the integral may be computed directly using the Cartan decomposition of $\mathrm{SL}_{2}(F)$ for $p$-adic fields $F/\mathbb{Q}_{p}$. See Remark 3.5 for the reason for the restriction to the irreducible case. The earlier work cited in the proof of Theorem 1.4 also amounts to the computation of an integral (1.2). After establishing several preliminary results, we prove Theorem 1.1 and its corollary in Section 3 below.

The algebraic group $\mathbf{G}_{\Delta}$ has a particularly complicated structure when $\Delta(x)$ is a power of a linear polynomial. The pro-isomorphic zeta functions of $\mathcal{L}_{\Delta}$ are obtained in [5] for $\Delta(x)=x^{2}$ and $\Delta(x)=x^{3}$ after computations substantially more involved than the ones in Section 3; it is notable that the simplifying assumptions used in [11] to analyze the integrals (1.2) do not hold in these cases.

This section mentions some results related to our work, as well as directions for future research.

We introduce a family of combinatorially defined functions in terms of which it will be convenient to express the local pro-isomorphic zeta functions $\zeta^{\wedge}_{\mathcal{L}_{f},p}(s)$. For every $r\in\mathbb{N}$, let $[r]$ denote the set $\{1,2,\dots,r\}$.

Let $p$ be a prime, and let $v_{p}$ be the normalized additive valuation on $\mathbb{Q}_{p}$. Let $F/\mathbb{Q}_{p}$ be a finite extension with ring of integers $\mathcal{O}_{F}$. Fix a uniformizer $\pi\in\mathcal{O}_{F}$. Let $k_{F}=\mathcal{O}_{F}/(\pi)$ be the residue field, and let $q$ denote its cardinality. Given $\lambda\in k_{F}^{\times}$, let $[\lambda]\in\mathcal{O}_{F}$ denote the $(q-1)$-st root of unity lifting $\lambda$, and set $[0]=0$. Let $I_{0}=\{0\}$, and for every $m\in\mathbb{N}$ define the set

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial of degree $n\geq 3$. Consider the primary polynomial $\Delta(x)=f(x)^{\ell}$ for $\ell\in\mathbb{N}$, and set $K_{\Delta}=\mathbb{Q}[x]/(\Delta(x))$; this ring has dimension $\ell n$ as a $\mathbb{Q}$-vector space. To describe the algebraic automorphism group of $L_{\Delta}$ we define three algebraic subgroups of $\mathbf{GL}_{2\ell n+2}$. There is a morphism of algebraic groups $\rho_{2}:\mathrm{Res}_{K_{\Delta}/\mathbb{Q}}\mathbf{SL}_{2}\to\mathbf{SL}_{2\ell n+2}$ given by

where for any $\mathbb{Q}$-algebra $R$ the map $\iota:K_{\Delta}\otimes_{\mathbb{Q}}R\to\mathrm{M}_{\ell n}(R)$ is determined by $\iota(\beta^{i}\otimes r)=rC_{\Delta}^{i}$ for any $i\in\mathbb{N}\cup\{0\}$ and $r\in R$, and $\beta=x+(\Delta(x))\in K_{\Delta}$. Equivalently, $\iota(\alpha)$ is the matrix of the $R$-linear endomorphism of $K_{\Delta}\otimes_{\mathbb{Q}}R$ corresponding to multiplication by $\alpha$, with respect to the basis $(\beta^{i}\otimes 1)_{i=0}^{\ell n-1}$. By a standard exercise in linear algebra, or [15, Theorem 1], the image of $\rho_{2}$ is indeed contained in $\mathbf{SL}_{2\ell n+2}$.

Consider the embedding of algebraic groups $\rho_{1}:\mathbf{G}_{m}\to\mathbf{GL}_{2\ell n+2}$ given by

and the embedding $\rho_{3}:\mathbf{G}_{a}^{4\ell n}\to\mathbf{SL}_{2\ell n+2}$ given by

Recall that our chosen $\mathbb{Z}$-basis of $\mathcal{L}_{\Delta}$ allows us to identify the algebraic automorphism group $\mathbf{G}_{\Delta}$ of $L_{\Delta}$ with an algebraic subgroup of $\mathbf{GL}_{2\ell n+2}$. Its structure, which was determined by Berman, Klopsch, and Onn, consists essentially of an internal semi-direct product of the three subgroups of $\mathbf{GL}_{2\ell n+2}$ just defined. There exists a symmetric matrix $\sigma\in\mathrm{GL}_{\ell n}(\mathbb{Z})$ such that $\sigma C_{\Delta}\sigma^{-1}=C_{\Delta}^{T}$; see [5, §2.1]. Set

From now on we assume $\ell=1$, namely that $\Delta(x)=f(x)\in\mathbb{Z}[x]$ is an irreducible monic polynomial of degree $n\geq 3$. To simplify the notation, write $\mathbf{G}\subset\mathbf{GL}_{2n+2}$ for $\mathbf{G}_{\Delta}$. Similarly, write $K$ for $K_{\Delta}$; this is the number field $\mathbb{Q}(\beta)$, where $\beta$ is a root of $f(x)$. Let $\mathcal{O}_{K}$ denote the ring of integers of $K$, and recall that the conductor $\mathcal{F}_{f}$ is the largest ideal of $\mathcal{O}_{K}$ contained in $\mathbb{Z}[\beta]$.

Now let $p$ be a rational prime that decomposes in $K$ as $p\mathcal{O}_{K}=\mathfrak{p}_{1}^{e_{1}}\cdots\mathfrak{p}_{r}^{e_{r}}$, where the distinct prime ideals $\mathfrak{p}_{i}\triangleleft\mathcal{O}_{K}$ have residue fields $\mathcal{O}_{K}/\mathfrak{p}_{i}$ of cardinality $q_{i}=p^{f_{i}}$. Then $\mathbb{Q}_{p}\otimes_{\mathbb{Q}}K\simeq F_{1}\times\cdots\times F_{r}$, where for every $i\in[r]$ we write $F_{i}$ for the localization $K_{\mathfrak{p}_{i}}$. Similarly, $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathcal{O}_{K}\simeq\mathcal{O}_{F_{1}}\times\cdots\times\mathcal{O}_{F_{r}}$.