Sous-groupes algébriques du groupe de Cremona

Pavage de Voronoï associé au groupe de Cremona

The action of the Cremona group of rank 2 on an infinite dimensional hyperbolic space is the main recent tool to study the Cremona group. Following the analogy with the action of PSL(2,Z) on the Poincaré half-plane, we exhibit a fundamental domain for this action by considering a Voronoï tessellation. Then we study adjacent cells to a given cell, as well as cells that share common points in the boundary at infinity.

Non simplicité du groupe de Cremona sur tout corps

Using a theorem of Dahmani, Guirardel and Osin we prove that the Cremona group in 2 dimension is not simple, over any field. More precisely, we show that some elements of this group satisfy a weakened WPD property which is equivalent in our particular context to the Bestvina and Fujiwara's one.

离散亨泽尔赋值环上Bruhat-Tits子群的算术

Arithmétique des sous-groupes de Bruhat-Tits sur un anneau de valuation discrète hensélien

本文细化了Bruhat-Tits子群的上同调研究，补充1987年工作，并对半单单连通群得到Grothendieck-Serre型结果及拟分裂伴随群情形的障碍精确计算。

关于与Cremona群相关的图的Gromov-双曲性

Sur l'hyperbolicité de graphes associés au groupe de Cremona

为强化映射类群与Cremona群的类比，研究Wright图及相关图的Gromov-双曲性，证明Wright图非双曲，构造两个图，一个拟等距于Wright图，另一个为双曲。

Partial canonical subgroups

The reduction of Siegel varieties modulo a prime number p is stratified by the multiplicative rank of the p-divisible group of the universal abelian variety. For r\geq 0 the maximal multiplicative subgroup of the restriction of the p-torsion group of the universal abelian variety to the r-th stratum lifts canonically to the tube of this stratum and defines a partial canonical subgroup of rank r. We prove that this subgroup extends in a finite flat way on some strict neighborhood of the tube. On the ordinary stratum and on its neighborhood, we recover the usual canonical subgroup considered by …

Finitude uniforme pour les cycles de codimension 2 sur les corps de nombres

Soit $X$ une variété projective et lisse, définie sur un corps de nombres. Sous l'hypothèse $H^2(X,\mathcal O_X)=0,$ Colliot-Thélène et Raskind ont démontré que le sous-groupe de torsion $CH^2(X)_{tors}$ du groupe de Chow en codimension $2$ est fini. Dans cette note, on donne des bornes uniformes pour le groupe fini $CH^2(X)_{tors}$ quand $X$ varie en famille. Let $X$ be a smooth projective variety defined over a number field. Assuming $H^2(X,\mathcal O_X)=0,$ Colliot-Thélène and Raskind proved that the torsion subgroup $CH^2(X)_{tors}$ in the Chow group of cycles of codimension $2$ is finite.…

Intersection de courbes et de sous-groupes, et problèmes de minoration de hauteur dans les variétés abéliennes C.M

We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the intersection of $X$ with the union of all subgroup schemes of codimension at least 2 is finite ; we settle the case where $A$ is a power of a simple Abelian variety of C.M. type. This generalises the previous known result, due to Viada and Rémond-Viada (who was able to prove the conjecture for power of an elliptic curve with complex multiplication). The proof is…

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble (Weak Approximation and Hasse Principle for homogeneous spaces with finite solvable stabilizer)

Soit $K$ un corps global et $G$ un $K$-groupe fini résoluble. Sous certaines hypothèses sur une extension déployant $G$, on démontre que l'espace homogène $V:=G'/G$ avec $G'$ un $K$-groupe semi-simple simplement connexe vérifie l'approximation faible. On utilise une version plus précise de ce résultat pour démontrer le principe de Hasse pour des espaces homogènes $X$ sous un $K$-groupe $G'$ semi-simple simplement connexe à stabilisateur géométrique $\bar G$ fini et résoluble, sous certaines hypothèses sur le $K$-lien $(\bar G,κ)$ défini par $X$. ----- Let $K$ be a global field and $G$ a finite…

Pure motives with representable Chow groups

Let $k$ be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over $k$ whose Chow groups are all representable belongs to the full and thick subcategory of motives generated by the twisted motives of curves. -- Motifs purs dont les groupes de Chow sont représentables. Soit $k$ un corps algébriquement clos. Nous prouvons, en nous servant de la théorie des motifs birationnels développée par Kahn et Sujatha, qu'un motif de Chow défini sur $k$ dont les groupes de Chow sont tous représentables appartient à la sous-catégorie pleine et ép…

Equivalence motivique des groupes algebriques semisimples

Two semisimple algebraic groups are said to be motivic equivalent if the motives of the associated twisted flag varieties are isomorphic modulo any prime p. The purpose of this note is to construct the combinatorial invariants which characterize motivic equivalence and which are the motivic analogues of the Tits indices which appear in the classification of semisimple algebraic groups. The expression of these invariants -the Tits p-indexes- in terms of the classical invariants associated to the natural underlying structures of semisimple algebraic groups allow to produce algebraic criteria of …

Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une variété propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des zé…

Théorèmes de dualité pour les corps de fonctions sur des corps locaux supérieurs et applications arithmétiques

Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a …

关于有理二次余弦周期

On rational quadratic cocycles

本文研究了有理二次空间的算术子群上同调中的循环类，并证明其生成函数为Siegel模形式。

Dualité et principe local-global sur des corps locaux de dimension 2

Let $k$ be an algebraically closed field, a finite field or a $p$-adic field. Let $K_0=k((x,y))$ be the field of Laurent series in two variables over $k$. We define Tate-Shafarevich groups of a commutative group scheme over $K_0$ via cohomology classes locally trivial at each completion of $K_0$ coming from a codimension 1 point of $\text{Spec}\; k[[x,y]]$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori. We apply these results to the study of the obstruction to the local-global principle for $K_0$-torsors under a connected linear algebraic …

Sur la cohomologie à support des fibrés en droites sur les variétés symétriques complètes

Let $X$ be a complete symmetric variety i.e. the wonderful compactification of a symmetric $G-$homogeneous space (where $G$ is a simply-connected semi-simple linear algebraic group). If $L$ is a line bundle over $X$ and if $C$ is a Bialynicki-Birula cell of codimension $c$ in $X$, then the Lie algebra $\mathfrak g$ of $G$ operates naturally on the cohomology group with support : $H^c_C(L)$. One gives here a necessary condition on the cell $C$ for that $\mathfrak g-$module have a finite dimensional simple subquotient. As applications one calculates the Euler-Poincaré characteristic of $L$ over …

Quelques questions d'approximation faible pour les tores algébriques

Let K be a global field, T a K-torus and S a finite set of places of K. Let K_v be the completion at a place v. Denote by T(O_v) the maximal compact subgroup of the group T(K_v) of K_v-points of T. We show that the diagonal map from T(K) to the product for v in S of the T(K_v)/T(O_v) need not be onto. As a corollary, for suitable v, the group T(O_v) does not cover all R-equivalence classes in T(K_v). While studying the height zeta functions of an algebraic torus over a function field in one variable over a finite field, D. Bourqui showed that Peyre's constant from the number field case should …