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

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

and we define a height

$\displaystyle 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

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

We make the following assumptions


For small enough $ U$,

Answers are positive if

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

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

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

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

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

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$,

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

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

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


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

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

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

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