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Jean-Louis Colliot-Thélène

Publications and source records attributed to Jean-Louis Colliot-Thélène.

At least 55 records · Page 3Linked to original sources
arXiv2014

Le complémentaire des puissances $n$-ièmes dans un corps de nombres est un ensemble diophantien

Given a number field $k$ and a positive integer $n$, there exists an algebraic variety $X$ over $k$ and a function $f$ on $X$ whose set of values $f(X(k))$ on the set of $k$-points of $X$ is the complement in $k$ of the set of $n$-th powers. This result had been proved by B. Poonen (2009) for $n$ a power of $2$. For $n$ arbitrary, under Schinzel's hypothesis, it has been given a conditional proof by T. Várilly-Alvarado and B. Viray (2012). Instead of Schinzel's hypothesis, we use "Salberger's trick", as developed in papers of Skorobogatov, Swinnerton-Dyer and one of the authors.

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arXiv2013

Rationalité d'un fibré en coniques

F. Campana had asked whether a certain threefold is rational. In arXiv:1310.3569v1 [mathAG], this variety was shown to be birational to a specific conic bundle and then to be unirational. We prove that this conic bundle is rational.

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arXiv2013

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 specialized group $G_x$ is isotropic over $k_v$. If the Brauer group of $X$ is reduced to the Brauer group of $k$, then strong approximation holds for $X$ away from the place $v$.

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arXiv2012

Groupe de Brauer non ramifié d'espaces homogènes de tores

Let k be a field, X a smooth, projective k-variety. If X is geometrically rational, there is an injective map from the quotient of Brauer groups Br(X)/Br(k) into the first Galois cohomology group of the lattice given by the geometric Picard group. In this note, where the main attention is on smooth compactifications of homogeneous spaces of algebraic k-tori, we show how under some hypotheses the map is onto, and how one may in some special case exhibit concrete generators in Br(X). This is applied to the analysis of counterexamples to the local-global principle for norms in biquadratic extensions of number fields.

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arXiv2012

Cycles de codimension 2 et H^3 non ramifié pour les variétés sur les corps finis

Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ implies a local-global principle of Brauer--Manin type for the Chow group of zero-cycles on $V$. This sheds a new light on work started thirty years ago. ----- Soit $X$ une variété projective et lisse sur un corps fini $\F$. On s'intéresse au groupe de cohomologie non ramifiée $H^3_\nr(X,\Q/\Z(2))$. Un faisceau de conjectures implique que ce groupe est fini. Pour certaines classes de solides, on a $H^3_\nr(X,\Q/\Z(2))=0$. Savoir si c'est le cas pour tout solide est un problème ouvert. Lorsqu'un solide $X$ est fibré au-dessus d'une courbe $C$, de fibre générique une surface projective et lisse $V$ sur le corps global $\F(C)$, la combinaison de $H^3_\nr(X,\Q/\Z(2))=0$ et de la conjecture de Tate pour $X$ a pour conséquence un principe local-global de type Brauer--Manin pour le groupe de Chow des zéro-cycles de la fibre générique $V$. Ceci éclaire d'un jour nouveau des investigations commencées il y a trente ans.

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arXiv2012

Groupe de Brauer non ramifié de quotients par un groupe fini

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 non ramifié du quotient SL(n)/G. On examine ce que donne sa méthode sur un corps k quelconque (de caractéristique nulle). Par cette méthode purement algébrique, on retrouve et étend des résultats obtenus par Harari et par Demarche au moyen de méthodes arithmétiques, comme la trivialité du groupe de Brauer non ramifié pour k= Q et G d'ordre impair.

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arXiv2011

Strong approximation for the total space of certain quadric fibrations

We study equations in four variables (x,y,z,t) of the shape q(x,y,z)=P(t), where q(x,y,z) is an indefinite ternary quadratic form over the integers and P(t) is a polynomial in one variable with integral coefficients. If P(t) is not the product of a constant and the square of a polynomial, strong approximation holds for integral solutions (x,y,z,t). In the general case, we show that the integral Brauer-Manin conditions are the only obstructions to strong approximation. We actually study the analogous situation over an arbitrary number field. --- Nous étudions les équations à quatre variables (x,y,z,t) à coefficients entiers du type q(x,y,z)=P(t), où q(x,y,z) est une forme quadratique entière ternaire indéfinie sur les réels, et P(t) un polynôme à coefficients entiers en une variable. Lorsque le polynôme n'est pas le produit d'une constante et d'un carré de polynôme, nous établissons l'approximation forte pour les solutions de ces équations en entiers (x,y,z,t). Dans le cas général, nous montrons que l'obstruction de Brauer-Manin entière est la seule obstruction à l'approximation forte. Nous étudions la situation sur un corps de nombres quelconque.

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arXiv2011

Quelques cas d'annulation du troisième groupe de cohomologie non ramifiée

The third unramified cohomology group is shown to vanish on certain varieties equipped with a pencil of quadrics or of smooth complete intersections of two quadrics. Over the complex field, this shows that the integral Hodge conjecture in degree 4 holds for such varieties. --- On établit la nullité du troisième groupe de cohomologie non ramifiée pour certaines variétés munies d'un pinceau de quadriques ou d'intersections complètes lisses de deux quadriques. Sur les complexes, ceci permet d'établir la conjecture de Hodge entière en degré 4 pour de telles variétés.

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arXiv2011

Sur le groupe de Brauer transcendant

For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. ---- Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur k un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient.

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arXiv2011

Cohomologie non ramifiée et conjecture de Hodge entière

Building upon the Bloch-Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the integral Hodge conjecture in degree 4 fails for unirational varieties of dimension at least 6. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fibre. En nous appuyant sur la conjecture de Bloch-Kato en K-théorie de Milnor, nous établissons un lien général entre le défaut de la conjecture de Hodge entière pour la cohomologie de degré 4 et le troisième groupe de cohomologie non ramifié à coefficients Q/Z. Ceci permet de montrer que sur un solide uniréglé le troisième groupe de cohomologie non ramifié à coefficients Q/Z s'annule, ce que la K-théorie algébrique ne permet d'obtenir que dans certains cas. Ceci permet à l'inverse de déduire d'exemples ayant leur source en K-théorie que la conjecture de Hodge entière pour la cohomologie de degré 4 peut être en défaut pour les variétés rationnellement connexes. Pour certaines familles à un paramètre de surfaces, on établit un lien entre la conjecture de Hodge entière et l'indice de la fibre générique.

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arXiv2010

Zero-cycles and rational points on some surfaces over a global function field

Let F be a finite field of characteristic p. We consider smooth surfaces over F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4 variables with coefficients in F, with d prime to p. We prove : For such surfaces over F(t), the Brauer-Manin obstruction to the existence of a zero-cycle of degree one is the only obstruction. For d=3 (cubic surfaces), this leads to the same result for rational points. -- Soit F un corps fini de caractéristique p. Pour une surface lisse sur F(t) définie par une équation f+tg=0, où f et g sont deux formes de degré d sur F en 4 variables, avec d premier à p, nous montrons que l'obstruction de Brauer-Manin au principe de Hasse pour les zéro-cycles de degré 1 est la seule obstruction. Pour d=3 (surfaces cubiques), on en déduit le même énoncé pour les points rationnels.

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arXiv2010

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$, and reduce the problem to the case where $G$ is simple. For simple groups we show that the answer is positive if $G$ is split of type ${\sf A}_{n}$ or ${\sf C}_n$, and negative for groups of other types, except possibly ${\sf G}_{2}$. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of $G$ on itself. The results and methods of this paper have played an important part in recent A. Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\sf A}_n$, ${\sf C}_n$, and ${\sf G}_2$.

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arXiv2009

Autour de la conjecture de Tate `a coefficients Z_l pour les vari'et'es sur les corps finis

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 closure of a finite field. In particular, we show that if the variety in question is a smooth projective complete intersection of dimension at least 3 and of degree prime to the characteristic, then a zero-cycle of degree one exists if the Tate conjecture is true for divisors on surfaces and the variety extends to a proper fibration over C all of whose fibres possess a component of multiplicity one.

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arXiv2007

Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points. ----- Une forme quadratique entière peut être représentée par une autre forme quadratique entière sur tous les anneaux d'entiers p-adiques et sur les réels, sans l'être sur les entiers. On en trouve de nombreux exemples dans la littérature. Nous montrons qu'une partie de ces exemples s'explique au moyen d'une obstruction de type Brauer-Manin pour les points entiers. Pour plusieurs types d'espaces homogènes de groupes algébriques linéaires, cette obstruction est la seule obstruction à l'existence d'un point entier.

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arXiv2006

Rational surfaces and canonical dimension of PGL_6

The "canonical dimension" of an algebraic group over a field by definition is the maximum of the canonical dimensions of principal homogenous spaces under that group. Over a field of characteristic zero, we prove that the canonical dimension of the projective linear group PGL_6 is 3. We give two distinct proofs, both of which rely on the birational classification of rational surfaces over a nonclosed field. One of the proofs involves taking a novel look at del Pezzo surfaces of degree 6.

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arXiv2005

Fields of cohomological dimension one versus C_1-fields

Ax gave examples of fields of cohomological dimension 1 which are not C_1-fields. Kato and Kuzumaki asked whether a weak form of the C_1-property holds for all fields of cohomological dimension 1 (existence of solutions in extensions of coprime degree rather than existence of a solution in the ground field). Using work of Merkur'ev and Suslin, and of Rost, D. Madore and I produced examples which show that the answer is in the negative. In the present note, I produce examples which require less work than the original ones. In the original paper, some of the examples were given by forms of degree 3 in 4 variables. Here, for an arbitrary prime p>3, I use forms of degree p in p+1 variables.

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