Algèbre et géométrie

We show that irreducibility of a polynomial in any number of variables over a number field is a diophantine condition, i.e. captured by an existential formula. This generalises a previous result by Colliot-Thélène and Van Geel that the set of non-nth-powers is diophantine for any n. Our method is heavily based on the Brauer group, originating from Poonen's use of quaternion algebras as a technical tool for first-order definitions in number fields.

Siegel proved the finiteness of the set of solutions to the unit equation in a number ring, i.e., for a number field K with ring of integers O, the equation x+y=1 has only finitely many solutions in O*. That is, reformulated in more algebro-geometric terms, the hyperbolic curve P^1-{0,1,infinite} has only finitely many 'integral points'. In 1983, Faltings proved the Mordell conjecture generalizing Siegel's theorem: a hyperbolic complex algebraic curve has only finitely many integral points. Inspired by Faltings's and Siegel's finiteness results, Lang and Vojta formulated a general finiteness […]

Gauss a donné des formules pour le nombre de points entiers primitifs de la 2-sphère de rayon au carré égal à n. Ces formules sont en termes de nombres de classes d'anneaux quadratiques de discriminant étroitement liés à n. Cela mène à la question de savoir si ceci peut être expliqué par une action libre et transitive du groupe de Picard de cet anneau sur l'ensemble des tels points entiers primitifs à symétries globales SO_3(Z) près. Ceci est en effet le cas, et cette action peut être explicitée. L'outil utilisé […]

Artin a résolu le 17ème problème de Hilbert en démontrant qu'un polynôme positif en n variables à coefficients réels est une somme de carrés de fractions rationnelles, et Pfister a montré que 2^n carrés suffisent. Dans cet exposé, on étudiera quand le théorème de Pfister peut être amélioré. On montrera qu'un polynôme réel positif de degré d en n variables est une somme de (2^n)-1 carrés si d<2n, et dans certains cas si d=2n.

I will explain some geometric ideas (mostly due to de Jong-Starr) one can use to study the Hasse principle for varieties defined over funvtion fields. I will illustrate these methods by giving a new proof of the classical result of Hasse -Minkowsky on quadrics.

It is well-known that weak approximation is birational invariant between smooth varieties by the implicit function theorem. For strong approximation, such property is no longer true. However one can expect that strong approximation is invariant between smooth varieties up to a closed sub-variety of codimension at least 2. Indeed, this result is proved for affine spaces in a joint work with Yang Cao which is applied to show strong approximation for toric varieties. Such result is also proved by Dasheng Wei by using a different method. In this talk, I'll […]