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

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.

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