Computing order zeta functions via resolution of singularities

Joshua Maglione

Universität Bielefeld

www.slides.com/joshmaglione/bva2019

jmaglione@math.uni-bielefeld.de

Creative Commons Attribution 4.0 International License

Enumerating subgroups of $\mathbb{Z}^d$

For $d=1$ , exactly one subgroup of index $n$ , for all $n \geq 1$ .

Record this in a Dirichlet generating function:

\displaystyle\zeta_{\mathbb{Z}}^{<}(s) = \sum_{H\leq_f \mathbb{Z}} |\mathbb{Z} : H|^{-s} = \sum_{n\geq 1} a_n(\mathbb{Z})n^{-s} = \sum_{n\geq 1} n^{-s}.

\displaystyle\zeta_{\mathbb{Z}}^{<}(s) = \sum_{H\leq_f \mathbb{Z}} |\mathbb{Z} : H|^{-s} = \sum_{n\geq 1} a_n(\mathbb{Z})n^{-s} = \sum_{n\geq 1} n^{-s}.

For the general case, it is slightly more complicated.

\displaystyle\zeta_{\mathbb{Z^d}}^{<}(s) = \zeta(s) \zeta(s - 1) \cdots \zeta(s - d + 1).

\displaystyle\zeta_{\mathbb{Z^d}}^{<}(s) = \zeta(s) \zeta(s - 1) \cdots \zeta(s - d + 1).

Challenging to explicitly compute for general groups, but $\mathbb{Z}^d$ is easy.

What about subrings of $\mathbb{Z}^d$ ?

All our subrings are unital. Nontrivial examples occur when $d \geq 3$ .

The $d = 3$ case follows from Datskovsky–Wright (1986).

Subring zeta function satisfies an Euler product decomposition:

\displaystyle\zeta_{\mathbb{Z}^d}(s) = \prod_{p \text{ prime}} \zeta_{\mathbb{Z}^d,p}(s) = \prod_{p \text{ prime}} \zeta_{\mathbb{Z}_p^d}(s).

\displaystyle\zeta_{\mathbb{Z}^d}(s) = \prod_{p \text{ prime}} \zeta_{\mathbb{Z}^d,p}(s) = \prod_{p \text{ prime}} \zeta_{\mathbb{Z}_p^d}(s).

\displaystyle\zeta_{\mathbb{Z}_p^3}(s) = \dfrac{1 + 2p^{-s} + p^{-2s}}{(1 - p^{-s}) (1 - p^{1 - 3s})}.

\displaystyle\zeta_{\mathbb{Z}_p^3}(s) = \dfrac{1 + 2p^{-s} + p^{-2s}}{(1 - p^{-s}) (1 - p^{1 - 3s})}.

The zeta functions $\zeta_{R, p}(s)$ are called local zeta functions.

Nakagawa (1996) and Liu (2007) prove that $\zeta_{\mathbb{Z}_p^4}(s)$ is

\dfrac{ 1 + 4p^{-s} + 2p^{-2s} + (4p - 3)p^{-3s} + \cdots + (3p^2 - 4p)p^{-6s} - 2p^{2-7s} - 4p^{2-8s} - p^{2-9s} }{ (1 - p^{-s})^2 (1 - p^{2-4s}) (1 - p^{3-6s})}

\dfrac{ 1 + 4p^{-s} + 2p^{-2s} + (4p - 3)p^{-3s} + \cdots + (3p^2 - 4p)p^{-6s} - 2p^{2-7s} - 4p^{2-8s} - p^{2-9s} }{ (1 - p^{-s})^2 (1 - p^{2-4s}) (1 - p^{3-6s})}

The $d \geq 5$ case is open.

Notice some properties of the local subring zeta functions:

Rational function in both $p$ and $p^{-s}$

$\Longrightarrow$ exists recurrence relation among $a_n(\mathbb{Z}^d)$

Palindromic numerators

$\Longrightarrow$ suggests restrictive geometric structure

Uniform for all primes $p$

$\Longrightarrow$ suggests nice structure of underlying varieties

\begin{aligned} \zeta_{\mathbb{Z}_p^3}(s) = \dfrac{1 + 2p^{-s} + p^{-2s}}{(1 - p^{-s}) (1 - p^{1 - 3s})} \end{aligned}

\begin{aligned} \zeta_{\mathbb{Z}_p^3}(s) = \dfrac{1 + 2p^{-s} + p^{-2s}}{(1 - p^{-s}) (1 - p^{1 - 3s})} \end{aligned}

The general setting

Fix a number field $K$ and its ring of integers $\mathcal{O}_K$ .

The order zeta function is

\displaystyle\zeta_K(s) = \sum_{1\in H\leq \mathcal{O}_K} |\mathcal{O}_K : H|^{-s} = \sum_{n\geq 1} a_n(\mathcal{O}_K) n^{-s},

\displaystyle\zeta_K(s) = \sum_{1\in H\leq \mathcal{O}_K} |\mathcal{O}_K : H|^{-s} = \sum_{n\geq 1} a_n(\mathcal{O}_K) n^{-s},

where $H$ runs over all finite index subrings of $\mathcal{O}_K$ , i.e. suborders.

These order zeta functions give analytic data of a fixed order, $\mathcal{O}_K$ .

Different perspectives consider all sub-orders with other properties.

See Kaplan et al. (2015), Brakenhoff (2009), Bhargava (2005).

Order zeta functions have an Euler product decomposition:

\displaystyle\zeta_K(s) = \prod_{p\text{ prime}} \zeta_{K, p}(s) = \prod_{p \text{ prime}} \zeta_{K\otimes\mathbb{Z}_p}(s),

\displaystyle\zeta_K(s) = \prod_{p\text{ prime}} \zeta_{K, p}(s) = \prod_{p \text{ prime}} \zeta_{K\otimes\mathbb{Z}_p}(s),

where the product runs over all rational primes $p$ in $K$ .

There are subtleties based on how $p$ splits in $K$ . We consider the totally split case, so that

\zeta_{K, p}(s) = \zeta_{\mathbb{Z}_p^d}(s)

\zeta_{K, p}(s) = \zeta_{\mathbb{Z}_p^d}(s)

The ring $\mathbb{Z}^d$ has one key advantage:

R \cong 1\!\cdot\!\mathbb{Z} \oplus R'.

R \cong 1\!\cdot\!\mathbb{Z} \oplus R'.

Instead, count multiplicatively closed sublattices of $R'$ :

\begin{aligned} \zeta_{\mathbb{Z}_p^d}(s) &= \zeta_{\mathbb{Z}_p^{d-1}}^{<}(s) \end{aligned}

\begin{aligned} \zeta_{\mathbb{Z}_p^d}(s) &= \zeta_{\mathbb{Z}_p^{d-1}}^{<}(s) \end{aligned}

That is, enumerate not necessarily unital subrings of $R'$ .

$p$ -adic cone integrals

We represent $\zeta_{K,p}^<(s)$ as a $p$ -adic cone integral. Notation:

For integers $\ell,m$ , a sequence of polynomials in $\mathbb{Z}[x_1,\dots, x_m]$ ,

\mathscr{D} = \left(f_0, g_0;\, f_1, g_1,\, \dots,\, f_\ell, g_\ell \right)

\mathscr{D} = \left(f_0, g_0;\, f_1, g_1,\, \dots,\, f_\ell, g_\ell \right)

is called cone integral data.

Associate to $\mathscr{D}$ , the closed subset of $\mathbb{Z}_p^m$ :

\mathscr{M}(\mathscr{D}, p) = \left\{x \in \mathbb{Z}_p^m ~\middle|~ \nu_p(f_i(x)) \leq \nu_p(g_i(x)), \; i\geq 1\right\}

\mathscr{M}(\mathscr{D}, p) = \left\{x \in \mathbb{Z}_p^m ~\middle|~ \nu_p(f_i(x)) \leq \nu_p(g_i(x)), \; i\geq 1\right\}

and the following cone integral:

Z_{\mathscr{D}}(s, p) = \displaystyle\int_{\mathscr{M}(\mathscr{D}, p)} \left|f_0(x)\right|_p^s \left|g_0(x)\right|_p \,d\mu(x).

Z_{\mathscr{D}}(s, p) = \displaystyle\int_{\mathscr{M}(\mathscr{D}, p)} \left|f_0(x)\right|_p^s \left|g_0(x)\right|_p \,d\mu(x).

\nu_p : p\text{-adic valuation},\qquad |\cdot |_p : p\text{-adic norm}, \\ \mu : \text{normalized Haar measure so }\mu(\mathbb{Z}_p^m) = 1.

\nu_p : p\text{-adic valuation},\qquad |\cdot |_p : p\text{-adic norm}, \\ \mu : \text{normalized Haar measure so }\mu(\mathbb{Z}_p^m) = 1.

The order zeta functions are equal to suitable cone integrals.

\displaystyle\zeta_{K, p}(s) = \int_{\mathscr{M}(\mathscr{D}, p)} |f_0(x)|_p^s |g_0(x)|_p \, d\mu(x).

\displaystyle\zeta_{K, p}(s) = \int_{\mathscr{M}(\mathscr{D}, p)} |f_0(x)|_p^s |g_0(x)|_p \, d\mu(x).

Ex: A cone integral for the subring zeta function of $\mathbb{Z}_p^3$ is

where the set $\mathscr{M}$ contains all $x\in\mathbb{Z}_p^3$ satisfying

\begin{aligned} \nu_p(x_{11}) &\leq \nu_p(x_{21}(x_{21} - x_{22})) .%, \\ %\nu_p(x_{11}) &\leq \nu_p(x_{21}(x_{31}-x_{32})), \\ %\nu_p(x_{22}) &\leq \nu_p(x_{32}(x_{32} - x_{33})), \\ %\nu_p(x_{11}x_{22}) &\leq \nu_p(x_{22}x_{31}(x_{31} - x_{33}) - x_{21}x_{32}(x_{32} - x_{33})) . \end{aligned}

Because the cone data is not (locally) monomial, computing this requires different tools.

\displaystyle \zeta_{\mathbb{Z}_p^3}(s+2) = (1 - p^{-1})^{-2} \int_{\mathscr{M}} |x_{11}x_{22}|_p^{s} |x_{22}|_p \, d\mu(x)

\displaystyle \zeta_{\mathbb{Z}_p^3}(s+2) = (1 - p^{-1})^{-2} \int_{\mathscr{M}} |x_{11}x_{22}|_p^{s} |x_{22}|_p \, d\mu(x)

Computing cone integrals via resolution of singularities

Theorem. (du Sautoy, Grunewald 2000)

Let $L$ be a ring. There are smooth algebraic varieties $V_t$ , $t\in\{1,\dots, m\}$ , defined over $\mathbb{Q}$ , and rational functions $W_t(X, Y)\in \mathbb{Q}(X, Y)$ , such that for almost all primes $p$ ,

\displaystyle\zeta_{L, p}^<(s) = \sum_{t=1}^m c_t(p) W_t(p, p^{-s}),

\displaystyle\zeta_{L, p}^<(s) = \sum_{t=1}^m c_t(p) W_t(p, p^{-s}),

where $c_t(p)$ denotes the number of $\mathbb{F}_p$ -rational points of $\overline{V}_t$ , the reduction of $V_t$ modulo $p$ .

From cone data $\mathscr{D} = (f_0, g_0;\, \dots,\, f_\ell, g_\ell)$ , we apply resolution of singularities to monomialize every $f_i$ and $g_i$ .

Want an explicit resolution of the ideal generated by

\displaystyle F_d(x) = \prod_{i=0}^\ell f_i(x)g_i(x).

\displaystyle F_d(x) = \prod_{i=0}^\ell f_i(x)g_i(x).

Challenges:

Number of variables for $\mathbb{Z}_p^d$ case is $\binom{d}{2}$ .

Little to no useful features of $F_d(x)$ in general.

\begin{aligned} F_3(x) &= (x_{21} - x_{22})x_{11}x_{21}x_{22} \\ F_4(x) &= (x_{22}x_{31}^2 - x_{21}x_{32}^2 + x_{21}x_{32}x_{33} - x_{22}x_{31}x_{33}) \\ &\quad \times (x_{32} - x_{33})(x_{31} - x_{32})x_{31}x_{32}x_{33}F_3(x) \end{aligned}

\begin{aligned} F_3(x) &= (x_{21} - x_{22})x_{11}x_{21}x_{22} \\ F_4(x) &= (x_{22}x_{31}^2 - x_{21}x_{32}^2 + x_{21}x_{32}x_{33} - x_{22}x_{31}x_{33}) \\ &\quad \times (x_{32} - x_{33})(x_{31} - x_{32})x_{31}x_{32}x_{33}F_3(x) \end{aligned}

A (sideways) tree of blowing up:

not locally monomial

locally monomial

Du Sautoy–Grunewald give blueprint to compute cone integrals by:

(1) applying a resolution of singularities and

(2) counting $\mathbb{F}_p$ -rational points on varieties.

Apply ideas of Bierstone–Milman (2006) and Blanco (2012a, 2012b) to monomialize binomials.

We need only monomialize $\mathscr{D}$ . Gives a little more freedom.

Tree of blowing ups for $\mathbb{Z}^4$

We have 74 charts, with 38 leaves and 185 monomial integrals.

We have verified $\zeta_{\mathbb{Z}_p^4}(s)$ with our methods.

From the du Sautoy, Grunewald theorem before,

\displaystyle \zeta_{K, p}(s) = \sum_{t=1}^m c_t(p) W_t(p, p^{-s}),

\displaystyle \zeta_{K, p}(s) = \sum_{t=1}^m c_t(p) W_t(p, p^{-s}),

where the $c_t(p)$ enumerated the $\mathbb{F}_p$ -rational points of $\overline{V}_t$ .

The Nakagawa, Liu formula for $\zeta_{\mathbb{Z}_p^4}(s)$ holds for all primes $p$ .

The varieties $V_t$ in our computation had "tame" structure.

For exactly 2 varieties, $c_t(p)$ depended on parity of $p$ .

Of the 185 varieties, computed $c_t(p)$ by hand for 13 of them.

For every variety, $c_t(p)$ is given as a polynomial in $p$ or 2 polynomials depending on if $p = 2$ or not.

An overview of the algorithm:

(1) Locally monomialize the cone data $\mathscr{D}$ ,

(2) Monomialize by counting $\mathbb{F}_p$ -rational points on varieties,

(3) Apply Barvinok's algorithm to evaluate monomial $p$ -adic integrals.

Still in a prototypical stage and shows promise:

for extending computations of $\zeta_{\mathbb{Z}^d}(s)$ and to other rings like $\mathbb{Z}[x]/(x^d)$ ,

for bringing computational tools from algebraic geometry to solve $p$ -adic integrals.