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.
Semi-stability and heights of cycles.
Séries d'Eisenstein et fonctions L de courbes elliptiques à multiplication complexe.
Shimura Varieties and Twisted Orbital Integrals.
Simultaneous approximation in number fields.
Solving algebraic equations in roots of unity.
Some abelian threefolds with nontrivial Griffiths group
Some observations on motivic cohomolgy of arithmetic schemes.
Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve.
Some simple elliptic surfaces of genus zero.
Some special curves of genus 5
Stably rational algebraic tori
The rationality of a stably rational torus with a cyclic splitting field is proved.
Stickelberger elements and modular parametrizations of elliptic curves.