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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 (Institut de Mathematiques de Jussieu)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

v1337

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.

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