In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface, the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called intersection tree by intersection theory. We construct a sequence of intersections, such that the multiplicity of a singular rational point is equal to that of one of the irreducible components in these intersections. The multiplicities of these irreducible components constructed above are bounded by their multiplicities in the intersection tree.
Let K be a complete discretely valued field with residue field k of characteristic p>0. There is a duality theory for cohomology with coefficients in commutative finite K-group schemes in the following cases : char(K)=0 and k finite (Tate), char(K)=p and k finite (Shatz), char(K)=0 and k algebraically closed (Bégueri). In this paper, we settle the case where char(K)=p and k is algebraically closed.
Using the construction of Colliot-Thélène and Ojanguren, we exhibit an example of a smooth projective geometrically rational variety X defined over a finite field F_p with an algebraic closure \bar F_p and the absolute Galois group G, such that the group H^3_nr(X, Z/2) is nonzero and the map CH^2(X)\to CH^2(\bar X)^G is not surjective.
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.…
Compter (rapidement) le nombre de solutions d'équations dans les corps finis
多项式方程组的有限域中的解的数目服从非常强的规律性，例如通过在有限域上定义的代数簇的zeta函数的合理性或在$\Q$上的椭圆曲线的Hasse-Weil的$L$ -函数的模性来反映。二十年来，已经发明了有效的方法来有效地计算这个数量的解决方案，特别是在密码学应用中。这篇文章介绍了其中的一些方法，一般依赖于使用莱夫谢茨的迹公式在一个适当的上同调理论，并讨论了各自的优点。- Le nombre de solutions dans les corps d'un système d'équations polynomiales obéit à une très forte régularité，especiétée par problem par la rationalité de la fonetzêta d'une variété algébrique sur un corps fini，or la modularité de la fonetzéta $L$ de Hasse-Weil d'une courbe elliptique sur $\Q$ . Depuis une vingtaine d'années des methodes efficaces ont été inventées pour calculer effectivement ce nombre de solut…
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…
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 …
This partly expository paper investigates versions of the Tate conjecture on the cycle map for varieties defined over finite fields with values in 'etale cohomology with Z_\ell-coefficients. The bulk of the paper is an exposition of a 1998 result of C. Schoen which shows that an integral version of the conjecture holds for 1-cycles provided the usual conjecture is true for divisors on surfaces. In a last section we then derive from Schoen's theorem new results on the existence of degree one zero-cycles on varieties defined over the function field of a smooth proper curve C over the algebraic c…
We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field $H$ of finite dimension over its center $h$. First, we show that solving a finite embedding problem over $H$ is equivalent to finding a solution to some finite embedding problem over $h$ fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions $H(t)$ with central indeterminate $t$ has a solution, if $h$ is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally…
We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any algebraically closed field extension, the degree map on the zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the Néron-Severi group has no torsion, we recover earlier results of A. Pirutka. The results rely on a detailed study of the third unramified cohomology group of specific products of varieties.
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 $k$ be a field of characteristic 0 and let $K$ be the function field of a smooth projective geometrically integral $k$-curve $X$. Let $T$ be a $K$-torus. In this article, we aim at studying the space of adelic points $T(S,\mathbb{A}_K)$ of $T$ outside a finite set $S$ of closed points of $X$. We start by proving that the group $T(K)$ of rational points of $T$ is always discrete (hence closed) in $T(S,\mathbb{A}_K)$. We then describe the quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ in each of the following three cases: $k$ is an algebraically closed field, $k$ is the field of Laurent series $\…
Let $X$ be a genus $d$ curve with $d\geq 2$ defined over a global function field $K$ of characteristic $p>0$ with $p>2d+1$. Suppose $X$ non-isotrivial. Let $Γ$ be a sub-group of $J(K_s)$, where $J$ is the jacobian of $X$ and $K_s$ is a separable closure of $K$, verifying $Γ/p Γ$ finite. Then one shows that $X\cap Γ$ has finite cardinal and one provides an explicit upper bound. This generalizes a result of Buium and Voloch.
Let X be an algebraic curve, defined over a perfect field, and G a divisor on X. If X has sufficiently many points, we show how to construct a divisor D on X such that l(2D-G)=0, of essentially any degree such that this is compatible the Riemann-Roch theorem. We also generalize this construction to the case of a finite number of constraints, l(k_i.D-G_i)=0, where |k_i|\leq 2. Such a result was previously claimed by Shparlinski-Tsfasman-Vladut, in relation with the Chudnovsky-Chudnovsky method for estimating the bilinear complexity of the multiplication in finite fields based on interpolation o…
The fundamental theorem of arithmetic factorizes any integer into a product of prime numbers. The Jordan-Holder theorem dissolves many groups by their normal series which can be refined into composition series. The main topic of this thesis is the dissociation of field extensions. We dissociate algebraic extensions by their intermediate fields to build a tower with the greatest possible number of Galois steps. We call "galtowerable" the extensions admitting a field tower all the steps of which are Galois extensions,i.e. a "Galois tower". Two Galois towers of the same galtowerable extension (fi…
In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let l be a prime number different from p. We develop a theory of l-modular types for the group GL(m,D), m\>1, in preparation of the study of the l-modular smooth representations of this group.
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.
Nous montrons que les corps p-adiques et les corps de nombres totalement imaginaires vérifient la propriété C_1^1 conjecturée par Kato et Kuzumaki en 1986. Autrement dit, si k est l'un de ces corps et f(x_0, ..., x_n) est un polynôme homogène de degré au plus n à coefficients dans k, tout élément de k s'écrit comme produit de normes depuis des extensions finies de k dans lesquelles f admet un zéro non trivial. Nous établissons aussi la conjecture d'Ax sur les corps pseudo-algébriquement clos parfaits pour les corps dont le groupe de Galois absolu est un pro-p-groupe. ----- We prove that p-adic…
In this article, we prove that if $H$ is a skew field of center $k$ and $σ$ an automorphism of finite order of $H$ such that the fixed subfield $k^{\langle σ\rangle}$ of $k$ under the action of $σ$ contains an ample field, then the inverse Galois problem has a positive answer over the skew field $H(t,σ)$ of twisted rational fractions. Moreover, if $k^{\langle σ\rangle}$ contains either a real closed field, or an Henselian field of residue characteristic $0$ and containing all roots of unity, then the profree group of countable rank $\widehat{F}_ω$ is a Galois group over $H(t,σ)$.
Modular symbols for the congruence subgroup $Γ_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform…
This is the first of a series of papers devoted to the stabilization of the twisted trace formula. It is just an introduction. We present the local theory of twisted endoscopy, following the fundamental works of Kottwitz-Shelstad, Labesse and Shelstad. Our presentation is partly new because we try to give canonical definitions of the basic objects, which are more usually defined using auxiliary data. We prove also some results about the image of endoscopic transfer and about the dual transfer of stable distributions whose support is a finite set of stable conjugacy classes.
Je retracerai l'histoire des conjectures de Weil sur le nombre de solutions d'équations polynomiales dans un corps fini et quelques unes des approches qui ont été proposées pour les résoudre. The Weil conjectures: origins, approaches, generalizations. I recount the history of the conjectures by Weil on the number of solutions of polynomial equations in finite fields, and some of the approaches that have been proposed to solve them.
Quelques questions d'approximation faible pour les tores algébriques
设K是整体域，T是K-环面，S是K的有限集.设K_v是在位置v上的完备化，T（O_v）表示T的K_v-点群T（K_v）的极大紧子群。本文证明了从T（K）到T（K_v）/T（O_v）的v在S中的乘积的对角映射不必是onto的。作为推论，对于适当的v，群T（O_v）不覆盖T（K_v）中的所有R-等价类。在研究有限域上单变量函数域上代数环面的高度zeta函数时，D.布尔基表明，佩尔的常数从数字领域的情况下，应乘以一个整数。为上述问题构造的同一类型的环面使我们能够证明这个整数不一定总是1。- Soient K un corps global，T un K-tore，S un ensemble fini de places de K. On note K_v le complété de K en une place v. Soit T（K），resp. T（K_v）分别是K-有理点群。K_v-rationnels，de T. Notons T（O_v）是T（K_v）的紧极大子群。我们认为T（K）的对角应用对T（K_v）/T（O_v）在S中的生成不是满射的。这意味着T（O_v）群的可转化性不需要强制T（K_v）的所有R-等价类。Lorsque K是一个函数的一个变量对一个函数的最后一个函数，在使用同一类型的tore时，我们的montrons que le facteur supplément…
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…
Admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le même résultat vaut pour n=4, c'est-à-dire pour les surfaces de del Pezzo de degré 4, lorsque le groupe de Brauer est réduit aux constantes et que la surface est suffisamment générale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics i…
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…
