Motivic Euler products in motivic statistics

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 memoir 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. It is a motivic analogue of Chambert-Loir and Tschinkel's work [CLT12] solving Manin's problem for integral points of bounded height on partial equivariant compactifications of vector groups over number fields, and a generalisation of Chambert-Loir and Loeser's paper [ChL]. Our main theorem, stated in the introduction and proved in the final chapter of the text, 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.Our method, based on the Poisson summation formula, is inspired by the one used in [CLT12], but the motivic setting requires several major adaptations besides the ones already present in [ChL]. The main contribution of this text is the introduction of a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, analogous to the classical Euler products encountered in number theory. It relies on the construction of geometric objects called symmetric products, which generalise the notion of symmetric power of a variety. These Euler products and symmetric products allow moreover a suitable extension of Hrushovski and Kazhdan's motivic Poisson formula, which is used to rewrite the motivic height zeta function as a sum, over characters, of series with coefficients in a Grothendieck ring of varieties with exponentials. The aforementioned weight topology on this ring is constructed using Saito's theory of mixed Hodge modules, as well as Denef and Loeser's motivic vanishing cycles and Thom-Sebastiani theorem.

Motivic Euler products in motivic statistics

“…where Y is a variety over a field K, pt i q iPI is a collection of indeterminates indexed by a set I, F P 1 `pt i q iPI K 0 pVar{Y qrrt i ss iPI , and F y ptq denotes the power series restricted to y P Y . Here we recall this definition, some properties, and prove some additional properties not contained in [2].…”

“…We overcome this difficulty by using the theory of motivic Euler products, as introduced by the first author [1,2]. Briefly, for X{C a variety and ta i u iPZě1 a set of classes in the relative Grothendieck ring K 0 pVar{Xq (or its localization M X :" K 0 pVar{XqrL ´1s), motivic Euler products give a systematic way to make sense of the expression ź xPX `1 `pa 1 q x t `pa 2 q x t 2 `.…”

“…In section 6 we summarize the construction of and basic results on motivic Euler products following [2], and provide some complements that will be necessary in the present work.…”

“…This question can be made precise by considering the probability for integers up to a fixed N , and then taking the limit as N Ñ 8. There is a simple heuristic which provides a candidate answer: an integer m is squarefree if and only if, for every prime number p, p 2 does not divide m. When N is large relative to p, the proportion of positive integers ă N that are divisible by p 2 is approximately 1{p 2 . If we assume these conditions are independent at different primes, we obtain the candidate probability ź p prime p1 ´1{p 2 q " ζp2q ´1…”

“…In this work, we remove that obstruction by using the theory of motivic Euler products developed by the first author in her work on motivic height zeta functions [1,2]. Combining this theory with a careful formalization of the motivic inclusion-exclusion method introduced by Vakil-Wood and some insights ported from probability theory using motivic random variables, we are able to prove motivic Bertini theorems that allow for very general Taylor conditions (cf.…”