Peyre: Motivic height zeta functions

Manin's principle in the functional case

The setting

Notation: let $p$ be a prime number, $q = p^k$, $\mathcal{C}$ a smooth projective curve over $\mathbb{F}_q$, $K = \mathbb{F}_q(\mathcal{C})$. A point $x \in \mathbb{P}_K^N$ induces a function

$$\tilde x : \mathcal{C} \rightarrow \mathbb{P}^N_{\mathbb{F}_q}$$

and we define a height

$$h_N(x) = \begin{cases}\deg (\tilde x^*(\mathcal{O}(1)) & \text{if $\tilde x$\ is not constant}\\ 0 & \text{otherwise} \end{cases}$$

Now let $V$ be a smooth, geometrically integral projective variety over $K$, let $U \subset V$ be an open subset, and define

$$Z_{U, k}(T) = \sum_{x \in U(k)} T^{h(x)}, \quad \zeta_{U, k}(s) = Z_{U, k} (q^{-s}).$$ (1)

We make the following assumptions

• $\omega_V^{-1}$ is very ample
• $\omega_V^{-1} = \phi^*(\mathcal{O}(1))$
• $H^1(V, \mathcal{O}_V) = 0$, $H^2(V, \mathcal{O}_V) = 0$
• $V(K)$ is Zariski dense

Problems

• Find the value of $\theta = \inf\{ \sigma \vert \zeta_{V,k}(s)$ converges if the real part $\Re(s) = \sigma\}$.
• Find the order of the pole of $\zeta_{U, k}$ at $\theta$.
• Find the leading coefficient of the Laurent series at $\theta$.

Questions
For small enough $U$,

• Is $\theta=1$?
• Is the order of $\zeta_{U,k}(s)$ at $\theta$ equal to $t = \rank{\mathrm{Pic}}(V)$?
• Is the leading term of $\zeta_{U,k}(s)$ at $\theta$ equal to $\theta_h^*(V)/(s-1)^t$, where

  $$\theta_h^*(V) = \alpha^*(V) w_h(\overline{V(K)})$$

  
  
  where $\alpha^*(V)$ can be defined in terms of the cone of effective divisors $C^1_{eff}(V) \subset {\mathrm{Pic}}V \tensor{\mathbb{R}}$, $w(h)$ is some adelic measure, and $\overline{V(K)}$ is the closure of the rational points?

Results
Answers are positive if

• $V = G/P$ where $G$ is a reductive group over $K$ and $P$ is a smooth parabolic subgroup of $G$ (Morris, EP).
• $V$ is a smooth toric projective variety (D. Bourgui), $U$ an open orbit of $V$.
• $V \subset \mathbb{P}_K^N$ is a hypersurface with $N >> \deg V$ (circle method).

Simplest example: $V = \mathbb{P}^n_K$. Let $g$ be the genus of $\mathcal{C}$.

$$\begin{multline*} Z_{V, k}(t) = q^{n(1-g)}\frac{\zeta_K((n+1)s - n)}{\zeta_K((... ...n+1)s - 1)}{\zeta_K((n+1)s)} + \frac{Q(q^{-s})}{\zeta((n+1)s)}. \end{multline*}$$

where $Q$ is a polynomial.

Motivic setting

Work in progress with A Chambert-Loir.

The ring of motivic integration

(Kontsevich, Denef, Loeser)

Definition: Let $k$ be a field, and let $\mathcal{M}_k$ be the ring with generators $[V]$ as $V$ ranges through varieties over $k$, subject to the relations $[V] = [V']$ if $V \iso V'$ and $[V] = [U] + [V-U]$, for $U$ open in $V$, and with multiplication given by the product of varieties.

(Note: De Jong pointed out some problem with this definition in positive characteristic.)

Now let $\mathbb{L} = [\mathbb{A}_k^1]$, $\mathcal{M}_{loc} = \mathcal{M}_k[\mathbb{L}^{-1}]$. Define a filtration by

$$F^m \mathcal{M}_{loc} = [V] \mathbb{L}^{-i} i - \dim V \geq m .$$

Let $\hat{\mathcal{M}} = \invlim \mathcal{M}_{loc}/F^m \mathcal{M}_{loc}$.

Motivic height
Let the notation be as in Section . Given an embedding $\phi: V \into \mathbb{P}^N_K$ we get a height $h:V(K) \rightarrow\mathbb{N}$. Given an open $U \subset V$, we can define varieties $U_n/k$ such that for all $k'/k$,

$$U_n(k') = \{x \in V(K')\vert h_{K'}(x) = n\},$$

where $K' = k'(\mathcal{C})$. Then define

$$Z_{U, k}(T) = \sum_{n \in \mathbb{N}} [U_n] T^n \in \mathcal{M}_k[[T]].$$

Examples:

• If $V$ is defined over $k$, $V_0 = V$, $V_n = \operatorname{Mor}_n(\mathcal{C}, V)$, morphisms of degree $n$
• If $V = \mathbb{P}_K^n$,

  $$Z_K(T) = \sum_{i \in \mathbb{N}}[\mathcal{C}^{(i)}] T^i$$

  
  
  where $\mathcal{C}^{(i)}$ is the $i$th symmetric product.

  $$Z_{\mathbb{P}_K^N, h}(T) = \mathbb{L}^{n(1-g)} \frac{Z_K(T^{n+1}\mathbb{L}^n)}{Z_K(T^{n+1})} + \frac{Q(T)}{Z_K(T^{n+1})}$$

  
  

  
  
  

  
  
  

  
  
  

Hope: generalize this to smooth cellular varieties over $k$.

Remark: Batyrev has a nice idea to attack this when $V$ is defined over $k$. But we have no idea what the relevant harmonic analysis is in this case.

A realization map
Suppose $k = \mathbb{F}_q$, and define a map $\mathcal{M}_k \rightarrow \mathbb{Z}$, $[V] \mapsto \char93 V(\mathbb{F}_q)$. Then we get a map

$$\mathcal{M}_{loc} \rightarrow \mathbb{Z}[q^{-1}]$$

$$\mathbb{L}^{-1} \mapsto q^{-1}$$

which takes $Z_{U, k}^{Mot}(T)$ to $Z_{U, k}^{funct}(T)$.

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