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

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

Compter (rapidement) le nombre de solutions d'\'equations dans les corps finis

The number of solutions in finite fields of a system of polynomial equations obeys a very strong regularity, reflected for example by the rationality of the zeta function of an algebraic variety defined over a finite field, or the modularity of Hasse-Weil's $L$-function of an elliptic curve over \Q. Since two decades, efficient methods have been invented to compute effectively this number of solutions, notably in view of cryptographic applications. This expos\'e presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discusses their respective advantages. ----- Le nombre de solutions dans les corps finis d'un syst\`eme d'\'equations polynomiales ob\'eit \`a une tr\`es forte r\'egularit\'e, refl\'et\'ee par exemple par la rationalit\'e de la fonction z\^eta d'une vari\'et\'e alg\'ebrique sur un corps fini, ou la modularit\'e de la fonction $L$ de Hasse-Weil d'une courbe elliptique sur \Q. Depuis une vingtaine d'ann\'ees des m\'ethodes efficaces ont \'et\'e invent\'ees pour calculer effectivement ce nombre de solutions, notamment en vue d'applications \`a la cryptographie. L'expos\'e en pr\'esentera quelques-unes, g\'en\'eralement fond\'ees l'utilisation de la formule des traces de Lefschetz dans une th\'eorie cohomologique convenable, et expliquera leurs avantages respectifs.Comment: S\'eminaire Bourbaki, 50e ann\'ee, expos\'e 968, Novembre 2006. 48 pages, in french. Final version to appear in Ast\'erisqu