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Hrushovski-Kazhdan's motivic Poisson formula and motivic height zeta functions

Model Theory in Geometry and Arithmetic May 12, 2014 - May 16, 2014

May 12, 2014 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Antoine Chambert-Loir (Université Paris-Sud (Orsay))
Location: MSRI: Simons Auditorium
Video

Abstract

I will describe joint work with François Loeser which was 

motivated by recent results with Yuri Tschinkel concerning the Manin conjecture about asymptotics of the number of points of bounded height in varieties over number fields.
Here, we investigate a geometric analogue. Namely,

we consider a projective variety $X$ over a projective curve $C$,

a normal-crossing divisor $D$ in $X$ such that $-(K_X+D)$ is big over the generic fiber. Given

a finite subset $S$ of $C$, we can then consider 

the moduli space $M_n$ of ``integral sections''  of $X\setminus D$ of degree $n$ with respect

to the log-anticanonical divisor $-(K_X+D)$ and form a generating series 

with coefficients in the Grothendieck group of varieties, the \emph{motivic height zeta function}.

Making use of Hrushovski-Kazhdan's motivic Poisson formula, we prove in some instances

($X\setminus D$ is a vector group, of which $X$ is an equivariant compactification)

of this setup that this series is rational, understand its denominator and its ``largest pole''.

As a consequence, we derive estimates for the dimension of $M_n$ and its number of irreducible components of maximal dimension.

Supplements
20723?type=thumb Chambert-Loir_Notes 607 KB application/pdf Download
Video/Audio Files

v1337

H.264 Video v1337.mp4 351 MB video/mp4 rtsp://videos.msri.org/data/000/020/629/original/v1337.mp4 Download
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