La méthode que Vojta a introduite dans sa preuve de la conjecture de Mordell et que Faltings a étendue pour prouver la conjecture de Lang sur les sous-variétés de variétés abéliennes repose sur une inégalité de hauteurs obtenue par approximation diophantienne. Nous montrons qu’une telle inégalité peut s’énoncer de manière très générale en dehors du contexte des groupes algébriques. Ce faisant, nous lui conférons également plus de souplesse, ce qui conduit à des applications nouvelles même sur les...

Consider an arbitrary algebraic curve defined over the field of all algebraic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such...

Let X be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\surd a,\surd b)/k}\left(x\right)=Q(t₁,...,tₘ)²$ over a number field k. We prove that: (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of X; (2) the analogue for rational points is also valid assuming Schinzel’s hypothesis.

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type ${\mathbf{A}}_4$.

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type ${\mathbf{A}}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.

Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion $X{{(}_k)}^{ét,Br}\subset X{{(}_k)}^{Br₁}$. In the first part, we apply ideas from the proof of $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if $\subset {\subset }_k$ are such that $\subset Ext\left({,}_k\right)$, then $X{{(}_k)}^{}=X{{(}_k)}^{}$. This allows us to conclude, among other things, that $X{{(}_k)}^{ét,Br}=X{{(}_k)}^{{}_k}$ and $X{{(}_k)}^{Sol,Br₁}=X{{(}_k)}^{So{l}_k}$.

Si un système d’équations polynomiales à coefficients entiers admet une solution dans ${𝐐}^n$, il en admet sur tout complété $p$-adique ou réel de $𝐐$. La réciproque a été démontrée par Hasse pour les quadriques, mais elle est fausse en général. Une grande partie des contre-exemples connus peuvent être expliqués à l’aide de l’obstruction de Brauer-Manin, basée sur la théorie du corps de classe. Il est donc naturel de se demander si, pour certaines classes de variétés, cette obstruction est la seule. Le but...

Let $\varphi :X\to X$ be a morphism of a variety defined over a number field $K$, let $V\subset X$ be a $K$-subvariety, and let ${𝒪}_{\varphi }\left(P\right)=\{{\varphi }^n\left(P\right):n\ge 0\}$ be the orbit of a point $P\in X\left(K\right)$. We describe a local-global principle for the intersection $V\cap {𝒪}_{\varphi }\left(P\right)$. This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of $V\left(K\right)$ are Brauer–Manin unobstructed for power maps on ${\mathbb{P}}^2$ in two cases: (1) $V$ is a translate of a torus. (2) $V$ is a line and $P$ has a preperiodic coordinate. A key tool in the proofs is the classical...

We present a collection of results on a conjecture of Jannsen about the p-adic realizations associated to Hecke characters over an imaginary quadratic field K of class number 1.The conjecture is easy to check for Galois groups purely of local type (Section 1). In Section 2 we define the p-adic realizations associated to Hecke characters over K. We prove the conjecture under a geometric regularity condition for the imaginary quadratic field K at p, which is related to the property that a global Galois...