### Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements

Joshua Maglione

Bielefeld University

math.uni-bielefeld.de/~jmaglione

www.slides.com/joshmaglione/cirm2021

## Igusa's zeta functions

Count roots of $$f\in\Z[X_1,\dots, X_d]$$: let $$p$$ be a prime,

N_m(p) = \#\left\{X\in(\Z/p^m\Z)^d ~\middle|~ f(X)\equiv 0\mod p^m\right\}
\displaystyle P_f(T) = \sum_{m=0}^\infty p^{-dm}N_m(p)T^m

Igusa's zeta function – $$p$$-adic integral closely related to

Of interest: $$f$$ a product of linear polynomials.

Goal: use combinatorial & topological tools to understand asymptotics and arithmetic of $$P_f(T)$$ and Igusa zeta function.

Comes up in computing:  subgroup, representation, & class counting zeta functions.

## Flag Hilbert–Poincaré series

$$\mathcal{A}$$ hyperplane arrangement – hyperplanes in $$K^d$$

$$\mathcal{L}(\mathcal{A})$$ intersection poset – all nonempty intersections

\pi_{\mathcal{A}}(Y) = \displaystyle\sum_{x\in\mathcal{L}(\mathcal{A})} \mu(x)(-Y)^{\mathrm{rk}(x)} = \mathrm{Poin}(\mathbb{C}^d \setminus \bigcup_{H\in\mathcal{A}}H;\; Y)

Poincaré polynomial:

$$K\!=\!\mathbb{C}$$

$$\hat{1}$$

$$\hat{0}$$

\mathcal{B}_n := \{X_i - X_j ~|~ 1 \leq i < j \leq n\}:
\pi_{\mathcal{B}_4}(Y) = 1 + 6Y + 11Y^2 + 6Y^3

$$\mathcal{L}(\mathcal{B}_4)$$

$$\mathrm{rk}$$

0

1

2

3

For $$F = (x_1 < \cdots < x_\ell)\in\Delta(\mathcal{L}(\mathcal{A}))$$,  with $$x_0=\hat{0}$$,  $$x_{\ell+1}=\varnothing$$,

\pi_F(Y) = \displaystyle\prod_{k=0}^{\ell} \pi_{\mathcal{A}^{x_k}_{x_{k+1}}}(Y)

With $$x\in \mathcal{L}(\mathcal{A})$$, two new arrangements

\begin{aligned} \mathcal{A}_x &= \left\{H\in\mathcal{A} ~\middle|~ x\subseteq H \right\}, & & (\text{subarrangement})\\ \mathcal{A}^x &= \left\{x\cap H ~\middle|~ H\in\mathcal{A}\setminus\mathcal{A}_x,\; x\cap H\neq \varnothing \right\}. & & (\text{restriction}) \end{aligned}

Example: $$\mathcal{A} = \mathcal{B}_4$$

Order complex: $$\Delta(\mathcal{L}(\mathcal{A}))$$ – flags in $$\mathcal{L}(\mathcal{A})$$

$$F$$ empty flag in $$\Delta(\mathcal{L}(\mathcal{A}))$$:

\begin{aligned} \pi_F(Y) &= \pi_{\mathcal{A}_{\varnothing}^{\hat{0}}}(Y) = \pi_{\mathcal{A}}(Y) \\ &= 1 + 6Y + 11Y^2 + 6Y^3 \end{aligned}

$$F = (\quad\;)$$:

\begin{aligned} \pi_F(Y) &= \pi_{\mathcal{A}^{\hat{0}}}(Y) \cdot\pi_{\mathcal{A}_{\varnothing}}(Y) \\ &= (1+Y)\cdot (1 + 3Y + 2Y^2) \end{aligned}
\begin{aligned} \pi_F(Y) &= \pi_{\mathcal{A}^{\hat{0}}}(Y) \cdot \pi_{\mathcal{A}_{\phantom{-}}}(Y) \cdot \pi_{\mathcal{A}_{\varnothing}}(Y) \\ &= (1+Y)\cdot (1 + Y) \cdot (1 + Y) \end{aligned}

$$F = (\quad\;<\;\quad)$$:

Set $$\widetilde{\mathcal{L}}(\mathcal{A}) = \mathcal{L}(\mathcal{A})\setminus\{\hat{0}\}$$ and $$\mathcal{A}$$ is central if $$\bigcap_{H\in\mathcal{A}} H \neq \varnothing$$

The flag Hilbert–Poincaré series:

\mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) \prod_{x\in F} \dfrac{T_x}{1 - T_x}

Thm (M.–Voll 2021). For $$\mathcal{A}$$ defined over characteristic $$0$$ and central, then

\mathsf{fHP}_{\mathcal{A}}\left(Y^{-1},(T_x^{-1})_{x\in\widetilde{\mathcal{L}}(\mathcal{A})}\right) = (-Y)^{-\mathrm{rk}(\mathcal{A})} T_{\hat{1}} \cdot \mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}).

## Self-reciprocity

Idea: $$\mathsf{fHP}_{\mathcal{A}}$$ is equivalent to a $$p$$-adic integral.

$$\mathcal{L}(\mathcal{B}_4)$$

Igusa zeta function: a substitution of $$\mathsf{fHP}_{\mathcal{A}}(Y, \bm{T})$$.

\begin{aligned} Y &= -p^{-1}, \\ T_x &= \begin{cases} p^{-1-s} & x = \phantom{-}, \\ p^{-2-2s} & x = \phantom{-}, \\ p^{-2-3s} & x = \phantom{-}, \\ p^{-3-6s} & x = \phantom{-}. \end{cases} \end{aligned}
\dfrac{(1-p^{-1})\left(1 - 5p^{-1} + 6p^{-2} + \cdots - 6p^{-4-5s} + 5p^{-5-5s} - p^{-6-5s}\right)}{(1 - p^{-1-s})^2(1 - p^{-2-3s})(1 - p^{-3-6s})}

Igusa zeta function of $$f_{\mathcal{B}_4}$$: for all primes $$p$$,

f_{\mathcal{B}_4}(\bm{X}) = (X_1 - X_2)(X_1 - X_3)(X_1 - X_4)(X_2 - X_3)(X_2 - X_4)(X_3 - X_4)

Class counting zeta function: a substitution of $$\mathsf{fHP}_{\mathcal{A}}(Y, \bm{T})$$.

\zeta^{\mathrm{cc}}_{\mathbf{G}}(s) = \displaystyle \sum_{m=0}^{\infty} \# \mathrm{cc}(\mathbf{G}(\mathbb{Z}/p^m\mathbb{Z})) p^{-ms}

$$\mathbf{G}$$ a nilpotent group scheme of finite type over $$\mathbb{Z}_p$$:

Set $$\Gamma = K_{3,2}$$ and $$\mathbf{G}_{\Gamma}$$ graphical group scheme.

$$\mathbf{G}_{\Gamma}(\mathbb{Z}_p)$$ =

$$\leq\mathrm{GL}_{12}(\mathbb{Z}_p)$$

$$=\langle I + E_{1,2} + E_{5,7} + E_{6,8},~\dots,$$

$$I + E_{1,6} + E_{2,8} + E_{3,10} + E_{4,12}\rangle$$

$$\mathcal{C}_n$$ coordinate hyperplanes in $$K^n$$ and $$\mathcal{L}(\mathcal{C}_n)\cong 2^{[n]}$$.

\begin{aligned} Y &= -p^{-1}, \\ T_I &= \begin{cases} p^{6-|I|-s} & I = \phantom{-}, \\ p^{7-|I|-s} & I = \phantom{-}, \\ p^{8-|I|-s} & I = \phantom{-}, \\ p^{4-s} & I = \phantom{-}, \\ p^{-|I|} & I = \phantom{-}. \end{cases} \end{aligned}

Class counting zeta function for $$\mathbf{G}_{\Gamma}(\mathbb{Z}_p)$$ for all primes $$p$$:

\dfrac{1 + p^{-s} - p^{1-s} - 2p^{2-s} - p^{3-s} + p^{4-s} + p^{4-2s}}{(1-p^{3-s})(1-p^{5-s})^2}

Details: Rossmann–Voll

## A different coarsening

Set each $$T_x=T$$ to get coarse flag Hilbert–Poincaré series:

Easier to see examples & other combinatorial properties.

Nice form:

\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \dfrac{\mathcal{N}_{\mathcal{A}}(Y, T)}{(1 - T)^{\mathrm{rk(\mathcal{A})}}}
\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) \left(\dfrac{T}{1 - T}\right)^{|F|}

Braid arrangement in $$\mathbb{Q}^n$$:

\mathcal{B}_n = \{X_i - X_j ~|~ 1\leq i < j \leq n\}
\mathcal{N}_{\mathcal{A}}(Y, T) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) T^{|F|}(1 - T)^{\mathrm{rk}(\mathcal{A})-|F|}
\mathcal{N}_{\mathcal{B}_2}(Y, T) = 1 + Y
\mathcal{N}_{\mathcal{B}_3}(Y, T) = 1 + 3Y + 2Y^2 + (2 + 3Y + Y^2)T
\begin{aligned} \mathcal{N}_{\mathcal{B}_4}(Y, T) &= 1 + 6Y + 11Y^2 + 6Y^3 \\ &\quad + (11 + 37Y + 37Y^2 + 11Y^3)T \\ &\quad + (6 + 11Y + 6Y^2 + Y^3)T^2 \end{aligned}

All nonnegative coefficients. Computed with $$\mathsf{HypIgu}$$.

\begin{aligned} \mathcal{N}_{\mathsf{E}_7}(Y, T) &= 1 + 63Y + 1617Y^2 + 21735Y^3 + 162939Y^4 + 663957Y^5 \\ &\quad + 1286963Y^6 + 765765Y^7 + (90400 + 1553980Y + 11064984Y^2 \\ &\quad + 42142884Y^3 + 92109360Y^4 + 113759940Y^5 + 70917656Y^6 \\ &\quad + 16725596Y^7)T + (5577043 + 64210477Y + 304475955Y^2 \\ &\quad + 763724661Y^3 + 1080226497Y^4 + 847444143Y^5 + 338480825Y^6 \\ &\quad + 53381039Y^7)T^2 + (37767356 + 323700436Y + 1123040604Y^2 \\ &\quad + 2022363924Y^3 + 2022363924Y^4 + 1123040604Y^5 \\ &\quad + 323700436Y^6 + 37767356Y^7)T^3 + (53381039 + 338480825Y \\ &\quad + 847444143Y^2 + 1080226497Y^3 + 763724661Y^4 + 304475955Y^5 \\ &\quad + 64210477Y^6 + 5577043Y^7)T^4 + (16725596 + 70917656Y \\ &\quad + 113759940Y^2 + 92109360Y^3 + 42142884Y^4 + 11064984Y^5 \\ &\quad + 1553980Y^6 + 90400Y^7)T^5 + (765765 + 1286963Y + 663957Y^2 \\ &\quad + 162939Y^3 + 21735Y^4 + 1617Y^5 + 63Y^6 + Y^7)T^6 \end{aligned}
\mathcal{N}_{\mathsf{E}_7}(1, T) = 2903040(1 + 120T + 1191T^2 + 2416T^3 + 1191T^4 + 120T^5 + T^6)

Do coefficients of $$\mathcal{N}_{\mathcal{A}}(Y, T)$$ have combinatorial interpretation?

## Coxeter arrangements

The Eulerian polynomial $$E_n(T)$$ is combinatorially defined.

\begin{array}{rcccccccccc} E_1: &&&&&& 1 &&&& \\ E_2: &&&&& 1 && 1 &&& \\ E_3: &&&& 1 && 4 && 1 && \\ E_4: &&& 1 && 11 && 11 && 1 & \\ E_5: && 1 && 26 && 66 && 26 && 1 \end{array}
\mathcal{N}_{\mathcal{A}}(1, T) = \pi_{\mathcal{A}}(1) \cdot E_{\mathrm{rk}(\mathcal{A})}(T).

Thm (M.–Voll 2021). For a Coxeter arrangement $$\mathcal{A}$$ with no $$\mathsf{E}_8$$ factors:

Conjecture (M.–Voll). For arbitrary $$\mathcal{A}$$, the coefficients of $$\mathcal{N}_{\mathcal{A}}(Y, T)$$ are nonnegative.

Apply combinatorial & topological tools from hyperplane arrangements to understand asymptotics and arithmetic of

\displaystyle P_f(T) = \sum_{m=0}^\infty p^{-dm}N_m(p)T^m.

Flag Hilbert–Poincaré series have combinatorial features:

Thm. $$\mathcal{A}$$ central and characteristic $$0$$ $$\implies$$ $$\mathsf{fHP}_{\mathcal{A}}$$ self-reciprocal.

Thm. $$\mathcal{A}$$ Coxeter and no $$\mathsf{E}_8$$-factor $$\implies$$ $$\mathcal{N}_{\mathcal{A}}(1, T)=\pi_{\mathcal{A}}(1)\cdot E_{\mathrm{rk}(\mathcal{A})}(T)$$.

$$\mathsf{fHP}_{\mathcal{A}}$$

Igusa

cc z.f.

$$\mathsf{cfHP}_{\mathcal{A}}$$

By Josh Maglione

# Flag Hilbert--Poincaré series and Igusa zeta functions of hyperplane arrangements

We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert--Poincaré series, and we show how these rational functions are connected to Igusa zeta functions. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics. This is joint work with Christopher Voll.

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