  A motivic height zeta function associated to a family of varieties
parametrised by a curve is the generating series of the classes, in the
Grothendieck ring of varieties, of moduli spaces of sections of this family
with varying degrees. This text is devoted to the study of the motivic height
zeta function associated to a family of varieties with generic fiber having the
structure of an equivariant compactification of a vector group. Our main
theorem describes the convergence of this motivic height zeta function with
respect to a topology on the Grothendieck ring of varieties coming from the
theory of weights in cohomology. We deduce from it the asymptotic behaviour, as
the degree goes to infinity, of a positive proportion of the coefficients of
the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get
an estimate for their dimension and the number of components of maximal
dimension. The main tools for this are a notion of motivic Euler product for
series with coefficients in the Grothendieck ring of varieties, an extension of
Hrushovski and Kazhdan's motivic Poisson summation formula, and a motivic
measure on the Grothendieck ring of varieties with exponentials constructed
using Denef and Loeser's motivic vanishing cycles.
