Heights and measures on analytic spaces. A survey of recent results, and some remarks

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.Comment: 41 pages, final version. To appear in: Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry, edited by Raf Cluckers, Johannes Nicaise, Julien Seba

Lectures on height zeta functions: At the confluence of algebraic geometry, algebraic number theory, and analysis

This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces

On the distribution of points of bounded height on equivariant compactifications of vector groups

We prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space. (Nous \'etablissons un d\'eveloppement asymptotique du nombre de points rationnels de hauteur born\'ee sur les compactifications \'equivariantes lisses de l'espace affine.)Comment: The theorem proven is a bit more general than in version 1. Clarification and reorganization of some parts of the pape

A non-archimedean Ax-Lindemann theorem

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.Comment: 31 pages; revised versio

Integral points of bounded height on partial equivariant compactifications of vector groups

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.Comment: 34 pages; revised version; submitte

Relations de Hodge--Riemann et combinatoire des matro\"ides (d'apr\`es K. Adiprasito, J. Huh et E. Katz)

Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincar\'e duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.Comment: S\'eminaire Bourbaki 2017/2018, 70e ann\'ee, expos\'e 1144, mars 2018. in French To appear in Ast\'erisque, vol. 41

Champs de Hurwitz

On construit les champs de Hurwitz et on en donne quelques propri\'et\'es, essentiellement contenues dans SGA 1. Quelques applications de nature arithm\'etique en sont d\'eduites. We propose a construction of Hurwitz stacks and give some properties of them, most of which are consequences of SGA 1. Some applications of arithmetic flavor are then deduced

Motivic height zeta functions

Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic fiber $X_F$ is a smooth equivariant compactification of $G$ such that $D=X_F\setminus G_F$ is a divisor with strict normal crossings, let $U$ be a surjective and flat model of $G$ over $C_0$. We consider a motivic height zeta function, a formal power series with coefficients in a suitable Grothendieck ring of varieties, which takes into account the spaces of sections $s$ of $X\to C$ of given degree with respect to (a model of) the log-anticanonical divisor $-K_{X_F}(D)$ such that $s(C_0)$ is contained in $U$. We prove that this power series is rational, that its "largest pole" is at $\mathbf L^{-1}$, the inverse of the class of the affine line in the Grothendieck ring, and compute the "order" of this pole as a sum of dimensions of various Clemens complexes at places of $C\setminus C_0$. This is a geometric analogue of a result over number fields by the first author and Yuri Tschinkel (Duke Math. J., 2012). The proof relies on the Poisson summation formula in motivic integration, established by Ehud Hrushovski and David Kazhdan (Moscow Math. J, 2009).Comment: 54 pages; revise