Motivic Euler products and motivic height zeta functions

TL;DR

This paper studies the motivic height zeta function for families of varieties with equivariant compactifications of vector groups, analyzing its convergence and asymptotic properties using motivic Euler products and advanced cohomological tools.

Contribution

It introduces a new framework for analyzing motivic height zeta functions, including convergence criteria and asymptotic behavior, using motivic Euler products and motivic Poisson summation.

Findings

Proves convergence of the motivic height zeta function in a weight topology.

Estimates the dimension and number of components of moduli spaces.

Describes asymptotic behavior of coefficients related to Hodge-Deligne polynomials.

Abstract

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.