Approximation forte pour les espaces homogènes de groupes semisimples sur le corps des fonctions d'une courbe algébrique complexe

复代数曲线函数域上的强逼近纯度

Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe

该研究证明复代数曲线函数域上，半单代数群齐性空间及低次仿射光滑完全交的余维2闭子集补集满足去掉非空有限位集的强逼近性质。

Courbes rationnelles sur les variétés homogènes et une désingularisation plus fine des variétés de Schubert

In this article we prove the irreducibility of the Hilbert scheme of rationnal curves on homogeneous varieties with fixed class in the Chow ring. This result has also been proved by J. F. Thomsen [T] and B. Kim and R. Pandharipande [KP]. Our method is totaly different (we don't use the compactification of stable maps) and enables us to prove the existence of rational smooth curves on homogeneous varities with fixed class in the Chow ring. This was not the case of Thomsen's and Kim and Pandharipande's proofs. We use a decomposition of G/P in orbits (called the P'-orbits, see definition) which a…

Dualité et principe local-global pour les tores sur une courbe au-dessus de C((t))

Pour $K$ un corps global (corps de nombres ou corps de fonctions d'une variable sur un corps fini $F$), on dispose de théorèmes de dualité classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne à valeurs dans des tores et des modules abéliens finis. Nous établissons de tels théorèmes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des séries formelles en une variable sur le corps des complexes. Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a ici des diff…

Approximation forte pour les variétés avec une action d'un groupe linéaire

Let $G$ be a connected linear algebraic group over a number field. Let $U \hookrightarrow X$ be a $G$-equivariant open embedding of a $G$-homogeneous space with connected stabilizers into a smooth $G$-variety. We prove that $X$ satisfies strong approximation with Brauer-Manin condition off a set $S$ of places of $k$ under either of the following hypotheses : (i) $S$ is the set of archimedean places; (ii) $S$ is a nonempty finite set and $\bar{k}^{\times}= \bar{k}[X]^{\times}$. The proof builds upon the case $X=U$, which has been the object of several works.

Approximation forte en famille

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the generic fibre of $f$ is a homogeneous space of a simply connected, almost simple, semisimple group $G/k(t)$, and that the geometric stabilizers are connected reductive groups. Let $v$ be a place of $k$ such that the fibration $f$ acquires a rational section over the completion $k_v$ at $v$. Assume moreover that at almost all points $x \in A^1(k_v)$ the special…

Compter (rapidement) le nombre de solutions d'équations 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 exposé presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discuss…