Motivic Euler Products and Motivic Height Zeta Functions

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