86 research outputs found

### 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

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