Ramifications on arithmetic schemes.
Rang de courbes elliptiques avec groupe de torsion non trivial
On construit des courbes elliptiques sur de rang au moins 3, avec un sous-groupe de torsion non trivial. Par spécialisation, des courbes elliptiques de rang 5 et 6 sur sont obtenues.
Rank frequencies for quadratic twists of elliptic curves.
Rational fixed points for linear group actions
We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group , an affine variety and a finite map , all defined over a finitely generated field of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set contains a Zariski dense sub-semigroup ; namely, there must exist an unramified covering and a map such that . In the case , is the additive group, we reobtain the...
Rational Points of Abelian Varieties with Values in Towers of Number Fields.
Rational points of bounded height on Fano varieties.
Rational points on curves
This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009.We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve over . The focus is on practical aspects of this problem in the case that the genus of is at least , and therefore the set of rational points is finite.
Rational points on elliptic curves over Q in elementary abelian 2-extensions of Q.
Rationale Punkte auf Fermatkurven und getwisteten Modulkurven.
Reelle algebraische Funktionen mit vorgegebenen Null- und Polstellen.
Refined theorems of the Birch and Swinnerton-Dyer type
In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.
Relations monomiales entre périodes p-adiques.
Representations of integers by binary forms and the rank of the Mordell-Weil group.