# Peyre: Motivic height zeta functions

Manin's principle in the functional case

The setting

Notation: let be a prime number, , a smooth projective curve over , . A point induces a function

and we define a height

Now let be a smooth, geometrically integral projective variety over , let be an open subset, and define

 (1)

We make the following assumptions
• is very ample
• ,
• is Zariski dense

Problems

• Find the value of converges if the real part .
• Find the order of the pole of at .
• Find the leading coefficient of the Laurent series at .

Questions
For small enough ,

• Is ?
• Is the order of at equal to ?
• Is the leading term of at equal to , where

where can be defined in terms of the cone of effective divisors , is some adelic measure, and is the closure of the rational points?

Results

• where is a reductive group over and is a smooth parabolic subgroup of (Morris, EP).
• is a smooth toric projective variety (D. Bourgui), an open orbit of .
• is a hypersurface with (circle method).

Simplest example: . Let be the genus of .

where is a polynomial.

Motivic setting

Work in progress with A Chambert-Loir.

The ring of motivic integration

(Kontsevich, Denef, Loeser)

Definition: Let be a field, and let be the ring with generators as ranges through varieties over , subject to the relations if and , for open in , and with multiplication given by the product of varieties.

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

Now let , . Define a filtration by

 subring generated by if

Let .

Motivic height
Let the notation be as in Section . Given an embedding we get a height . Given an open , we can define varieties such that for all ,

where . Then define

Examples:
• If is defined over , , , morphisms of degree
• If ,

where is the th symmetric product.

Hope: generalize this to smooth cellular varieties over .

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

A realization map
Suppose , and define a map , . Then we get a map

which takes to .

Back to the main index for Rational and integral points on higher dimensional varieties.