The primitive curves are the multiple curves that can be locally embedded in smooth surfaces (we will always suppose that the associated reduced curves are smooth). These curves have been defined and studied by C. Banica and O.Forster in 1984. In 1995, D. Bayer and D. Eisenbud gave a complete description of the double curves. We give here a parametrization of primitive curves of arbitrary multiplicity. Let Z_n=spec(\C[t]/(t^n)). The curves of multiplicity n are obtained by taking an open cover (U_i) of a smooth curve C and by glueing schemes of type U_i X Z_n using automorphisms of U_{ij} X Z_n that leave U_{ij} invariant. This leads to the study of the sheaf of nonabelian groups G_n of automorphisms of C X Z_n that leave the reduced curve invariant, and to the study of its first cohomology set. We prove also that in most cases it is the same to extend a primitive curve C_n of multiplicity n to one of multiplicity n+1, and to extend the quasi locally free sheaf D_n of derivations of C_n to a rank 2 vector bundle on C_n.
Familles de polynomes liees aux courbes modulaires X_1(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient
因此，一个完整的和E_{c,l}一个家庭的库伯特的课程椭圆定义上Q munies d 'un point rationnel A d' ordre l。关于F_{c,l}，我们注意到E_{c,l}对A的生成群的椭圆商，以及E_{c,l}对F_{c,l}的同构.对于l = 3，4，5和6，我们将F_{c,l}（Q）/（f_l（E_{c,l}（Q）的非平凡元素c的参数化表示为显式表示，F_{c,l}（Q）的显式点不是E_{c,l}（Q）的无平凡元素f_l的图像。这些点是无限大的。我们将此方法应用于构造Q度的扩展环，并通过柄部和Gras恢复某些钝体。在另一篇文章中，我们研究了在此获得的某些扩展的专有算法。
Let K be the function field of a curve over the complex field. Let X be a homogeneous space of a semisimple linear algebraic group. Strong approximation holds for X outside any finite nonempty set of places of K. Strong approximation fails for tori over K.
The moduli space of stable curves of Deligne and Mumford is a compactification of the moduli space of smooth curves of genus >=2 that parametrizes certain nodal curves. It is a powerful tool for the study of algebraic curves. Higher-dimensional analogues were constructed by Kollár, Shepherd-Barron and Alexeev in dimension 2, and by Viehweg in the case of smooth varieties. We will explain the recent ideas allowing for the construction of these moduli spaces in general, including the stable reduction theorem in higher dimension, which reflects their compactness. L'espace de modules des courbes s…
Let X be a smooth projective curve over a field of characteristic p>0 and G a finite group of automorphism of X. Let n(X,G) be the characteristic of the versal equivariant deformation ring R(X,G) of (X,G). When the ramification is weak (i.e. all second ramification groups are trivial),we prove that n(X,G) is 0 or p and we compute R(X,G).
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque
设X是数域k上的光滑真簇. Colliot-Thélène、Kato和Saito提出了关于X的零圈Chow群在对应群在k的所有完备上的乘积中的像的猜想。在奇异纤维的交换性假设下，我们证明了在有限Tate-Shafarevich群曲线上纤维化的全空间上，这些结构为满足弱逼近的有理连通簇. - Soit X une variété propre et lisse sur un corps de nombres k.本文介绍了Colliot-Thélène、Kato和Saito提出的关于X的零周期的周星驰群的图像。我们对各种合理连接的总纤维空间的分析表明，近似值是不可能的，因此，Tate-Shafarevich集团的课程结束了，我们对单一纤维的假设进行了分析。
Groupe de Picard de la variét\e de modules des thêta-caracteristiques des courbes planes
我们计算了$d$级课程平面（非强制列表）的物理特性模块$Θ_{\planp}(d)$的Picard空间群。我们对$d\ge 6$的认识是，Picard的组合是由${\cal{D}}$和${\cal{L}}$组成的，在形态学支持下，${\cal{O}}_{\proj}(1)$或$\proj$是$d$对$\planp$的程度课程的空间。对于$1\le d \le 4$，该小组是由${\cal{L}}$创建的，在这些情况下，确定性很小。在韦伊分部的$\Cl$小组中，确定性的人承认一个种族：${\cal{P}}$的候选人。此外，对于$d\geq 4$对，${\cal{L}}$还在$\Cl$中加入了一个外消旋体。如果是$d=5$，则将其转换为一个可逆矩阵，Picard的群由${\cal{L}}$和${\cal{P}}$生成。On verra ainsi que la composante paire de $Θ_{\planp}(d)$ est localement factoriel si et seulement si $d=1,2,3$ or $d=5$ .结果是一个类似的结果。接着，我们研究了一个普遍存在的家庭的问题，即$U$的一个特征不太稳定的问题。我们认为$d$对一个家庭不存在，但$d$损害一个家庭存在，而不是在整个$U$上，但在Zombski的拓扑学中是局部的。
Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (…
Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$. We give a formula for the number of irreducible $\ell$-adic local systems ($\ell\neq p$) with a fixed rank over $X$ fixed by the Frobenius endomorphism. We prove that this number behaves like a Lefschetz fixed point formula for a variety over $\mathbb{F}_q$, which generalises a result of Drinfeld in rank $2$ and proves a conjecture of Deligne. To do this, we pass to the automo…
L1：我们继续对G. Castelnuovo关于使亏格>1的曲线的每一点固定的复平面上的双有理变换群，我们象Castelnuovo那样使用曲线的伴随线性系。我们证明了这些群是阿贝尔群，并且这些群要么是有限的，阶数为2或3，要么共轭于德琼基耶尔群的子群。我们还表明，这些结果不推广到曲线的亏格<=1。- 法语：Nous poursuivons l 'étude faite par G. Castelnuovo en 1892 au sujet du groupe des transformations birationnelles du plan Schwarzenburg qui fixent points par points une courbe de genre >1; nous nous servons comme lui des systèmes linéaires adjoints de la courbe birth.我们认为这两个小组都是阿贝里昂人，而且他们都是二阶或三阶的，因此与琼基耶尔小组的一个小组相结合。我们的montrons平等地认为，这些结果不具有一般性，而不属于<=1的类型。
In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by André, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language o…
Dualité et principe local-global pour les tores sur une courbe au-dessus de C((t))
为了$K$一个全球性的团体（团体的nombres或团体的职能d 'une变量sur un corps fini $F$），在处理对偶分类理论（泰特，普瓦图，中山），为一个cohomologie galoisienne à valeurs dans des tores和des modules abéliens dans。我们为$K$建立了一套理论体系，其中包含一个射影函数，并在复体上的一个变量上的$F={\bf C}((t))$系列形式的函数上建立了一套理论体系。Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a dictionary avec le cas classique（$K$ corps global），and aussi avec le cas récement étudié oeconomy $K$ est un corps de functions d'une variable sur un corps $p$ -adique.另一方面，$K$ --一种不隐含的合理性并不意味着哈塞原理对于空间的原理是一致的。在整体域$K$（数域，或有限域$F$上的曲线的函数域）上…
This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the étale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or noda…
We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation $ψ$ of the maximal stable state $μ\equiv 3$ is obtained by adding extra grains at several points. It appears, that the result $ψ^\circ$ of the relaxation of $ψ$ coincides with $μ$ almost everywhere; the set where $ψ^\circ\ne μ$ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points. Nous considérons le modèle du tas de sable sur l'ensemble des points entiers d'un polygone entier. En aj…
We prove that, over any elliptic global Langlands parameter $σ$, the cuspidal cohomology groups of moduli stacks of shtukas are given by a formula involving a finite dimensional representation of the centralizer of $σ$. It is a first step in the direction of Arthur-Kottwitz conjectures.
