Manin's principle in the functional case
Notation: let be a prime number, , a smooth projective curve over , . A point induces a function
Now let be a smooth, geometrically integral projective variety over , let be an open subset, and define
For small enough ,
Answers are positive if
Simplest example: . Let be the genus of .
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 the notation be as in Section . Given an embedding we get a height . Given an open , we can define varieties such that for all ,
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
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