Apparaît dans la collection : Model Theory and Valued Fields

Analogously to the height zeta function linked to Manin’s problem of counting rational points of bounded height on varieties, we consider the motivic height zeta function that enumerates moduli spaces of sections of varying degree of a given family of varieties over a curve; it takes its coefficients in a Grothendieck ring of varieties. I will explain the results of Margaret Bilu’s PhD thesis in which she describes the behaviour of this motivic height zeta function in the case where the generic fiber of this family is an equivariant compactification of a vector group. In particular, she shows that a positive proportion of the top part of the Hodge-Deligne polynomial of these moduli spaces behaves as if the motivic height zeta function were a rational power series. Her results rely on a construction of motivic Euler products and a generalization of Hrushovski-Kazhdan’s motivic Poisson formula. They require to complete the Grothendieck ring of varieties with exponentials with respect to a weight topology which is defined using total vanishing cycles and the Thom Sebastiani formula.