# Computing order zeta functions via resolution of singularities

### Joshua Maglione

Universität Bielefeld

jmaglione@math.uni-bielefeld.de

Reporting on work in progress with ...

Universität Bielefeld

Universität Oldenburg

Universität Oldenburg

## 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}.

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).

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}_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})}

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}

## 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},

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.

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),

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)

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

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}

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)

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\}

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).
\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).

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)

## 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}),

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).

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}

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.

## Solving monomial integrals

Once $$\mathscr{D}$$ is monomial, translate $$p$$-adic integral to counting integral points on rational polyhedron $$\mathcal{P}\subseteq \mathbb{R}^r$$.

Theorem. (Barvinok (1994))

There exists an algorithm that, given a rational polyhedron $$\mathcal{P}$$ of a fixed dimension $$r\in\mathbb{N}$$, returns the generating function in poly-time

Barvinok's algorithm has implementations in Sage via LattE.

We use Rossmann's Zeta package to compute monomial integrals, which also employs LattE, Rossmann (2018).

\displaystyle F_{\mathcal{P}}(X) = \sum_{\alpha\in \mathcal{P}\cap \mathbb{Z}^r} X^\alpha.

## 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}),

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.

By Josh Maglione

# Computing order zeta functions via resolution of singularities

For a number field K with ring of integers O, the order zeta function of K is a Dirichlet generating series enumerating orders, i.e. unital subrings of O of finite index. In comparison with the Dedekind zeta function of K, the order zeta function of K is poorly understood: for number fields of degree larger than 5, next to nothing general is known. Encoding this Dirichlet series as a p-adic integral, we develop computational tools to repeatedly resolve singularities until it is distilled to enumerating points on polyhedra and p-rational points of algebraic varieties. This is joint work with Anne Fruehbis-Krueger, Bernd Schober, and Christopher Voll.

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