*Manin's principle in the functional case*

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

We make the following assumptions

- is very ample
- ,
- is Zariski dense

- 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 .

- 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*

Answers are positive if

- 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.

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.