HrushovskiKazhdan's motivic Poisson formula and motivic height zeta functions
Model Theory in Geometry and Arithmetic May 12, 2014  May 16, 2014
Location: SLMath: Eisenbud Auditorium
v1337
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 normalcrossing 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 loganticanonical 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 HrushovskiKazhdan'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.
ChambertLoir_Notes

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