Gaël Rémond

Sur les sous-variétés des tores

In Invent. Math.126 (2000), pp. 513–545, we gave a proof of Langs conjecture on Abelian varieties leading to an effective bound for the number of translates involved. We show here that the method can be extended to give a similar statement for the ‘Mordell–Lang plus Bogomolov’ theorem proven by B. Poonen and independently by S. Zhang. We deal in detail with tori for which effective results have been obtained by J.-H. Evertse and H. P. Schlickewei; we improve on these mainly by providing polynomial bounds in the degree instead of doubly exponential ones. We also state a theorem for Abelian varieties. In both cases the strategy of proof is based on the approach of Mumford and Vojta–Faltings–Bombieri together with an effective Bogomolov property and therefore does not rely on either equidistribution nor subspace theorem arguments.

Théorème des périodes et degrés minimaux d'isogénies

We give a new, sharpened version of the period theorem of Masser and Wustholz, which is moreover totally explicit. We also present a new formulation involving all archimedean places. We then derive new bounds for elliptic isogenies, improving those of Pellarin. The small numerical constants obtained allow an application to Serres uniformity problem in the split Cartan case, thanks to the work of Bilu, Parent and Rebolledo.

Polarisations et isogénies

Soit A une variete abelienne definie sur un corps de nombres k. Nous demontrons qu’il existe un faisceau inversible ample et symetrique sur A dont le degre est borne par une constante explicite qui depend seulement de la dimension de A, de sa hauteur de Faltings et du degre du corps de nombres k. Nous etablissons egalement des versions explicites du theoreme de Bertrand relatif au theoreme de reductibilite de Poincare et des theoremes d’isogenies de Masser et Wustholz entre varietes abeliennes. Les preuves reposent sur des arguments de geometrie des nombres dans les reseaux euclidiens constitues des morphismes entre varietes abeliennes munis des metriques de Rosati. Nous majorons les minima successifs de ces reseaux grâce au theoreme des periodes que nous avons demontre dans un article precedent.

Problème de Mordell-Lang modulo certaines sous-variétés abéliennes

Following a result of Bombieri, Masser and Zannier on tori, the second author proved that the intersection of a transversal curve C in a power A of a C. M. elliptic curve with the union of all algebraic subgroups of Eg of codimension 2 is finite. Here transversal means that C is not contained in any translate of an algebraic subgroup of codimension 1. We merge this result with Faltings’ theorem that C ∩ Γ is finite when Γ is a finite rank subgroup of A. We obtain the finiteness of the intersection of C with the union of all Γ + B for B an abelian subvariety of codimension 2. As a corollary, we generalize the previous result to a curve C not contained in any proper algebraic subgroup, but possibly contained in a translate. We also have weaker analog results in the non C. M. case.

INTERSECTION DE SOUS-GROUPES ET DE SOUS-VARIÉTÉS II

We elucidate the structure of various exceptional subsets appearing in parts I and II in order to prove new results on the Zilber–Pink conjecture for abelian varieties. In particular, we obtain boundedness of height on the intersection of interest for all non-degenerate varieties (in a precise sense). The main idea to prove that our exceptional subsets are closed comes from a result of Bombieri, Masser and Zannier on tori and we follow the same approach through so-called anomalous subvarieties but we have to allow extra generality in the definition. We also use in a crucial way a theorem of Ax on analytic subgroups of algebraic groups. Mathematics Subject Classification (2000). 11G10, 11G50, 14K12. Mots-Clefs. Variétés abéliennes, conjecture de Zilber–Pink.

Espaces adéliques quadratiques

We study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K , we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.

Variétés abéliennes et ordres maximaux

We prove that an abelian variety whose endomorphism ring is a maximal order can be written as a direct product of simple factors with the same property, in which furthermore two isogenous factors have isomorphic nth powers for some n. Conversely every such product has a maximal order as endomorphism ring. We deduce from this some properties for arbitrary abelian varieties, in particular for almost complements of abelian subvarieties.

Une Inégalité de Łojasiewicz Arithmétique

Une inegalite de Łojasiewicz minore la valeur |f(x)| d’une fonction analytique f : ℝn → ℝ par une puissance de la distance de x a l’ensemble des zeros de f. Nous nous interessons ici au cas arithmetique ou f est un polynome a coefficients entiers.