We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\to G$, all defined over a finitely generated field $\kappa$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi \left(V\right(\kappa \left)\right)$ contains a Zariski dense sub-semigroup $\Gamma \subset G\left(\kappa \right)$; namely, there must exist an unramified covering $p:\tilde{G}\to G$ and a map $\theta :\tilde{G}\to V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb{Q}$, $G={𝔾}_a$ is the additive group, we reobtain the...

Nous complétons l’interprétation géométrique de [P2]1991:726;1401792m:11061 pour les hauteurs locales archimédiennes et les distances projectives de [P1]88h11048. On montre comment ceci conduit à une taille (telle que définie dans [P3]) sur les anneaux de coordonnées de variétés projectives. On définit aussi des notions de maison et taille pour les extensions de type fini de $\mathbf{Q}$.

Soit $f:𝒳\to C$ une surface projective fibrée au-dessus d’une courbe et définie sur un corps de nombres $k$. Nous donnons une interprétation du rang du groupe de Mordell-Weil sur $k\left(C\right)$ de la jacobienne de la fibre générique (modulo la partie constante) en termes de moyenne des traces de Frobenius sur les fibres de $f$. L’énoncé fournit une réinterprétation de la conjecture de Tate pour la surface $𝒳$ et généralise des résultats de Nagao, Rosen-Silverman et Wazir.

We study the family of curves $F_m\left(p\right):x^p+y^p=m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_m\left(p\right)$. As a corollary we conclude that the jacobians of the curves $F_m\left(5\right)$ with even analytic rank and those with odd analytic rank are equally distributed.

We construct del Pezzo surfaces of degree $4$ violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of $K3$ surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.

Let $X$ be an algebraic variety defined over a field $k$ of characteristic $0$, and let $Y$ be an $X$-torsor under a torus. We compute the Brauer group of $Y$. In the case of a number field $k$ we deduce results concerning the arithmetic of $X$.

Let $K$ be a number field, and let $A/K$ be an abelian variety. Let $c$ denote the product of the Tamagawa numbers of $A/K$, and let $A{\left(K\right)}_{\mathrm{tors}}$ denote the finite torsion subgroup of $A\left(K\right)$. The quotient ${c/|A\left(K\right)}_{\mathrm{tors}}|$ is a factor appearing in the leading term of the $L$-function of $A/K$ in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over $\mathbb{Q}$ or quadratic extensions $K/\mathbb{Q}$, and for abelian surfaces $A/\mathbb{Q}$. The smallest possible ratio...