Programme des cours

Semaine 1 — Cours d’introduction

Christophe Soulé - Arithmetic intersection

Résumé :
  Let \(X\) be a 2-dimensional, normal, flat, proper scheme over the integers. Assume \(\bar L\) and \(\bar M\) are two hermitian line bundles over \(X\). Arakelov (and Deligne) defined a real number \(\bar L . \bar M\), the arithmetic intersection number of \(\bar L\) and \(\bar M\). We shall explain the definition and the basic properties of this number. Next, we shall see how to extend this construction to higher dimension, and how to interpret it in terms of arithmetic Chow groups.

Jean-Benoît Bost - Theta series, infinite rank Hermitian vector bundles, and Diophantine algebraization

Résumé :
  In the classical analogy between number fields and function fields, an Euclidean lattice \((E, \Vert .\Vert)\) may be seen as the counterpart of a vector bundle \(V\) on a smooth projective curve \(C\) over some field \(k\). Then the arithmetic counterpart of the dimension \(h^0(C, V) = \dim_k \Gamma(C,V)\) of the space of sections of \(V\) is the non-negative real number $$h^0_{\theta}(E, \Vert.\Vert) := \log \sum_{v \in E} e^{- \pi \Vert v \Vert^2}.$$ In these lectures, I will firstly discuss diverse properties of the invariant \(h^0_{\theta}\) and of its extensions to certain infinite dimensional generalizations of Euclidean lattices. Then I will present applications of this formalism to transcendence theory and to algebraization theorems in Diophantine geometry.

Éric Gaudron - Minima et pentes des espaces adéliques rigides

Résumé :
  Ce cours présente un abrégé de la théorie des minima et pentes successives des espaces adéliques rigides sur une extension algébrique du corps des nombres rationnels. Seront réunis dans un même tout une partie de la géométrie des nombres des ellipsoïdes de Minkowski, la théorie des pentes des fibrés vectoriels hermitiens de Bost et le formalisme des hauteurs tordues de Roy et Thunder.

Huayi Chen - Théorème de Hilbert-Samuel arithmétique

Résumé :
  Le théorème de Hilbert-Samuel en géométrie algébrique relie le comportement asymptotique du système linéaire gradué d’un faisceau inversible ample au nombre d’intersection. Gillet et Soulé ont démontré un analogue arithmétique de ce résultat. Dans ce mini-cours, j’explique cet énoncé arithmétique et l’idée de sa démonstration.

Per Salberger - Quantitative aspects of rational points on algebraic varieties

Résumé :
  Let \(X\) be a subvariety of \(\mathbf P^n\) defined over a number field and \(N(B)\) be the number of rational points of height at most \(B\) on \(X\). There are then general conjectures of Manin on the asymptotic behaviour of \(N(B)\) when \(B\) goes to infinity. These conjectures can be studied using the Hardy-Littlewood method for non-singular complete intersections of high dimensions and by adelic harmonic analysis for varieties related to algebraic groups. But for most varieties there are no other methods available apart from sieve theory and determinant methods. The latter was first developed for affine plane curves in a paper of Bombieri-Pila and extended to projective varieties by Heath-Brown and myself. The goal of the \(p\)-adic version of this method is to show that the set of rational points of height at most \(B\) on \(X\) in a congruence class satisfy further equations if \(p\) is large enough compared to \(B\). One can then proceed by induction with respect to dim\((X)\) to obtain uniform upper estimates for \(N(B)\) like a proof of the dimension growth conjecture of Heath-Brown and Serre.
  The lectures will focus on the determinant method. We will make essential use of Mumford’s geometric invariant theory too see how stability conditions affect the equidistribution of rational points. This is inspired by Donaldson’s and Tian’s theory of Kähler-Einstein metrics. We will also explain how one can use the theory of volumes of line bundles in Lazarsfeld’s book to improve upon the original bounds of Heath-Brown. If time permits we will also mention a number of striking similarities between the determinant method and methods used in the theory of Diophantine approximation.


Semaine 2 - Equidistribution et dynamique

Emmanuel Peyre - Slopes and distribution of points

Résumé :
  The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These lectures will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arise when one considers rational points of bounded height near a fixed rational point.

Antoine Chambert-Loir - Equidistribution theorems in Arakelov geometry and Bogomolov conjecture

Résumé :
  Let \(X\) be an algebraic curve of genus \(g\geqslant 2\) embedded in its Jacobian variety \(J\). The Manin-Mumford conjecture (proved by Raynaud) asserts that \(X\) contains only finitely many points of finite order. When \(X\) is defined over a number field, Bogomolov conjectured a refinement of this statement, namely that except for those finitely many points of finite order, the Néron-Tate heights of the algebraic points of \(X\) admit a strictly positive lower bound. This conjecture has been proved by Ullmo, and an extension to all subvarieties of Abelian varieties has been proved by Zhang soon after. These proofs use an equidistribution theorem in Arakelov geometry due to Szpiro, Ullmo, and Zhang. Using more classical techniques of diophantine geometry, David and Philippon have given another proof which, moreover, provides an effective lower bound.
  In the talk, I will present the equidistribution statement, and his powerful generalization due to Yuan. I will then give the proof of the Bogomolov conjecture following Ullmo-Zhang. If time permits, I will also describe the proof of David and Philippon. I then plan to introduce the non-archimedean analogue of the equidistribution result and its application by Gubler to the Bogomolov conjecture over function fields.

Pascal Autissier - Autour du théorème de Fekete-Szegö

Résumé :
  On s'intéresse dans ce cours à la situation suivante:
  Soit \(\mathcal K\) un compact de \(\mathbb{C}\). En théorie du potentiel, on associe à \(\mathcal K\) sa capacité \(\gamma\), qui est un réel positif. Ce nombre, défini de manière analytique, mesure en quelque sorte la "taille arithmétique" de \(\cal K\). C'est ce qu'illustre le théorème de Fekete et Szegö:
  Pour \(U\subset\mathbb{C}\), notons ici \(\mathcal Y(U)\) l'ensemble des \(\alpha\in\overline{\mathbb{Z}}\) tels que \(\alpha\) et tous ses conjugués (par Galois) soient dans \(U\).

Théorème (Fekete, Szegö). Soient \(\mathcal K\) un compact de \(\mathbb{C}\) symétrique par rapport à l'axe réel. Désignons par \(\gamma\) sa capacité.

  • Si \(\gamma\lt 1\), il existe un ouvert \(U\) de \(\mathbb{C}\) contenant \(\mathcal K\) tel que \(\mathcal Y(U)\) soit fini.
  • Si \(\gamma\geqslant1\), pour tout ouvert \(U\) de \(\mathbb{C}\) contenant \({\mathcal K}\), \({\mathcal Y}(U)\) est infini.

La théorie du potentiel se généralise en fait aux surfaces de Riemann compactes. On expliquera comment ceci permet d'étendre la situation précédente au cas des surfaces arithmétiques. Enfin, dans le cas critique \(\gamma=1\), on montrera une généralisation du théorème d'équidistribution de Bilu.

Romain Dujardin - Some problems of arithmetic origin in complex dynamics and geometry

Résumé :
  Some themes inspired from number theory have been playing an important role in holomorphic and algebraic dynamics (iteration of rational mappings) in the past ten years. In these lectures I would like to present a few recent results in this direction. This should include:

  • the dynamical Manin-Mumford problem, in particular in the case of product rational maps \((P(x),Q(y))\) (after Ghioca, Nguyen, and Ye)
  • the “unlikely intersection” problem (after Baker and DeMarco, and also Favre and Gauthier).

A key technical tool in these results is the equidistribution theory of points of small height. If time permits, we’ll also discuss the related problem of the equidistribution of roots of random polynomials.


Semaine 3 - Variétés de Shimura

Jan Bruinier et José Ignacio Burgos Gil - Arakelov theory on Shimura varieties

Résumé :
   A Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over \(\mathbf Q\). Shimura varieties have a very rich geometric and arithmetic structure. For instance they are defined over a number field (the reflex field), they have line bundles provided with hermitian metrics that come from a representation of a maximal compact subgroup and sometimes they have models over a localization of a ring of integers coming from a modular interpretation.
   Open Shimura varieties admit toroidal compactifications, but the mentioned metrized line bundles do not extend to a smoothly metrized line bundle in the compactification, but to a line bundles with logarithmic singular metric. Thus the usual Arakelov geometry can not be applied to them. In this course we will explain how to extend Arakelov theory to cover this class of singular metrics.
   Important applications of this extended Arakelov theory arise in the context of the Kudla program, which predicts deep connections between the arithmetic geometry of arithmetic special cycles on integral models of orthogonal and unitary Shimura varieties and the theory of Siegel modular forms. These connections lead to (often conjectural) generalizations of results of Gross, Kohnen and Zagier on Heegner divisors on modular curves. We will give an introduction to the Kudla program and discuss some cases where the predictions have been proved.

Gerard Freixas i Montplet - Automorphic forms and arithmetic intersections

Résumé :
  In these lectures I will focus on the Riemann-Roch theorem in Arakelov geometry, in the specific context of some simple Shimura varieties. For suitable data, the cohomological part of the theorem affords an interpretation in terms of both holomorphic and non-holomorphic modular forms. The formula relates these to arithmetic intersection numbers, that can sometimes be evaluated through variants of the first Kroenecker limit formula. I will first explain these facts, and then show how the Jacquet-Langlands correspondence allows to relate arithmetic intersection numbers for different Shimura varieties, whose associated groups are closely related.

Fabrizio Andreatta - The height of CM points on orthogonal Shimura varieties and Colmez conjecture

Résumé :
  We will first introduce Shimura varieties of orthogonal type, their Heegner divisors and some special points, called CM (Complex Multiplication) points. Secondly we will review conjectures of Bruinier-Yang and Buinier-Kudla-Yang which provide explicit formulas for the arithmetic intersection of such divisors and the CM points. We will show that they imply an averaged version of a conjecture of Colmez. Finally we will present the main ingredients in the proof of the conjectures.
   The lectures are base on joint works with E. Goren, B. Howard and K. Madapusi Pera.






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