Colliot-Th{é}l{è}ne has determined the Chow group of zero-cycles on a Ch{â}telet surface X defined over a finite extension K of the field of p-adic numbers (p an odd prime) when X is split by an unramified extension of K. Using similar techniques, we extend this calculation to the ramified case, simplifying his result in the process. We also determine how this group changes when we pass from K to a finite extension of K.
We determine the Chow group of zero-cycles on a rational surface X defined over a finite extension K of the field of p-adic numbers (p a prime) when X is split by an unramified extension of K.
Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.
For a homogeneous space X of a connected algebraic group G (with connected stabilizers) over a field k of characteristic zero, we construct a canonical complex of Galois modules of length 3 and a canonical isomorphism between an hypercohomology group of this complex and an explicit subgroup of the Brauer group of X. This result is obtained as a consequence of a formula describing the "algebraic Brauer group of a torsor", and it generalizes recent results by Borovoi and van Hamel, by considering non-linear groups G and by taking into account some transcendental elements in the Brauer group of X…
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.
Given a projective variety X over an algebraically closed field of characteristic zero, we show that finite parabolic bundles along a fixed simple normal crossings divisor D are in one to one correspondence with representations of the étale fundamental group of X-D.
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.
We are interested in the actions of an algebraic group G over the grassmannians of a finite dimensional K-vector space V (K algebraically closed) deduced from an action of G over V. We prove that the dimension of the orbit space of G(i,V) is smaller than that of G(j,V) if and only if the dimension of G(i,V) is smaller than that of G(j,V). We then get similar results for flag varieties. Different methods allow us to get results about the number of orbits when the ground field is finite.
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…
We prove that the arithmetic fundamental group of X admits no section over the absolute Galois group of Q when X is the Schinzel curve, thereby confirming in this example the prediction given by Grothendieck's section conjecture. ----- Nous démontrons que le groupe fondamental arithmétique de X n'admet pas de section au-dessus du groupe de Galois absolu de Q lorsque X est la courbe de Schinzel, confirmant ainsi sur cet exemple la prédiction donnée par la conjecture des sections de Grothendieck.
On montre que le groupe de Selmer d'une isogénie de hauteur un entre deux variétés abéliennes définies sur le corps de fonctions d'une variété quasi-projective et lisse $V$ sur un corps parfait $k_0$ de caractéristique $p>0$ peut être plongé dans le groupe des homomorphismes entre deux fibrés vectoriels naturels sur $V$. / We show that the Selmer group of an isogeny of height one between two abelian varieties defined on the function field of a smooth and quasi-projective variety $V$ over a perfect field $k_0$ of characteristic $p>0$ can be embedded in the group of homomorphisms between two nat…
Nous calculons le groupe de Picard de l'espace des modules $Θ_{\planp}(d)$ des thêta-caractéristiques des courbes planes (non forcément lisses) de degré $d$. Nous montrons que pour $d\ge 6$, le groupe de Picard de la composante paire est engendré par le fibré déterminant ${\cal{D}}$ et l'image réciproque ${\cal{L}}$, sous le morphisme support schématique, de ${\cal{O}}_{\proj}(1)$ où $\proj$ est l'espace des courbes de degré $d$ sur $\planp$. Pour $1\le d \le 4$, ce groupe est engendré uniquement par ${\cal{L}}$, le fibré déterminant étant trivial dans ces cas. Dans le groupe $\Cl$ des diviseu…
Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of t…
Let k be a field, G a finite group embedded in the k-group SL(n). For k an algebraically closed field, Bogomolov gave a formula for the unramified Brauer group of the quotient SL(n)/G. We develop his method over any characteristic zero field. This purely algebraic method enables us to recover and generalize results of Harari and of Demarche over number fields, such as the triviality of the unramified Brauer group for k=Q and G of odd order. --- Soient k un corps et G un groupe fini plongé dans le k-groupe SL(n).Pour k algébriquement clos, Bogomolov a donné une formule pour le groupe de Brauer …
This is a thoroughly revised version of math.AG/0502516v1 (24 Feb. 2005). Let k be a field of characteristic zero. Let Y=G/H, where G is a connected linear algebraic group over k and H is a connected closed k-subgroup of G. Let X be a smooth compactification of Y over k. We prove the conjecture set forward in the previous version : the Galois-lattice given by the geometric Picard group of X is flasque. The previous version had only partial results in this direction. They were obtained at the expense of a long d'etour via local and global fields. We owe the drastic improvement to a suggestion b…
Soit K le corps des fractions d'une surface projective et lisse, géométriquement intègre, définie sur un corps fini F. Soit C/K une conique. Parimala et Suresh ont montré que le groupe de cohomologie non ramifiée en degré trois de K(C) à coefficients de torsion est nul. Dans cette note on étend leur résultat aux variétés de Severi-Brauer associées à une algèbre centrale simple dont d'indice l est premier et différent de char F.
(On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je démontre que le groupe fondamental d'une variété normale rationnellement connexe par cha\^ınes est fini. La démonstration est valable en caractéristique différente de zéro.
Le texte présente les fondements d'une théorie du groupe fondamental en Géométrie Algébrique, dans le point de vue ``kroneckerien'' permettant de traiter sur le même pied le cas d'une variété algébrique au sens habituel, et celui d'un anneau des entiers d'un corps de nombres, par exemple. The text presents the foundations of a theory of the fundamental group in Algebraic Geometry from the Kronecker point of view, allowing one to treat on an equal footing the case of an algebraic variety in the usual sense, and that of the ring of integers in a number field, for instance.
Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Thélène, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the i…
this paper is devoted to the study of curves of genus 3 with group of automorphisms the symmetric group S3, principally over finite fields, in view to obtain optimal curves. For instance, we prove that, over the finite fields of char. 3, for any elliptic curve E with j-invariant not in F_3, there exists a curve of genus 3 whose jacobian is isogeneous to E^3; we have also results in char. 7, and these curves provide many examples of optimal curves in any char. Nous etudions les courbes de genre 3 dont le groupe d'automorphismes est S3, en vue d'obtenir des courbes optimales sur les corps finis.…
Étant donné un anneau de valuation $V$, de corps résiduel $F$ et de groupe des valeurs $Γ$, on donne une condition suffisante pour qu'un anneau local dominant $V$ soit un anneau de valuation de groupe $Γ$. Lorsque $V$ contient un corps $k$, ce résultat est appliqué à la construction d'un anneau de valuation contenant $V$ et une extension donnée $k'$ de $k$, de groupe $Γ$ et de corps résiduel engendré par $k'$ et $F$. Cela s'avère possible, notamment, lorsque $k'$ ou $F$ est séparable sur $k$. Given a valuation ring $V$, with residue field $F$ and value group $Γ$, we give a sufficient condition…
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…
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.…
En utilisant un théorème de Gabber sur les altérations, on démontre un résultat décrivant la partie de torsion première à $p$ du groupe de Brauer non ramifié d'une variété $V$ lisse et géométriquement intègre sur un corps global de caractéristique $p$ au moyen de l'évaluation des éléments de $\Br V$ sur ses points locaux. Using a theorem of Gabber on alterations, we prove a result describing the prime-to-$p$ torsion part of the unramified Brauer group of a smooth and geometrically integral variety $V$ over a global field of characteristic $p$ by evaluating the elements of $\Br V$ at its local …
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 …
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 …
Let A be a complex abelian variety and G its Mumford--Tate group. Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank of G, which is a little less than log_2 dim A. If we suppose that End A is commutative, then we show that rk G >= log_2 dim A + 2, and this latter bound is sharp. We also obtain the same results for the rank of the l-adic monodromy group of an abelian variety defined over a number field. ----- Soit A une variété abélienne complexe et G son groupe de Mumford--Tate. En supposant que les sous variétés abéliennes simple…
English: We continue the study of G. Castelnuovo on the group of birational transformation of the complex plane that fix each point of a curve of genus >1; we use adjoint linear system of the curve as Castelnuovo does. We prove that these groups are abelian, and that these are either finite, of order 2 or 3, or conjuguate to a subgroup of the de Jonquières group. We show also that these results do not generalise to curves of genus <=1. ----- French: Nous poursuivons l'étude faite par G. Castelnuovo en 1892 au sujet du groupe des transformations birationnelles du plan complexe qui fixent points…
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 …
Nous montrons, pour une grande famille de propriétés $P$ des espaces homogènes, que $P$ vaut pour tout espace homogène d'un groupe linéaire connexe dès qu'elle vaut pour les espaces homogènes de $\mathrm{SL}_n$ à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d'une importante conjecture de Colliot-Thélène sur l'obstruction de Brauer-Manin au principe de Hasse et à l'approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s'applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-cycles. We pr…
