Soit $N$ un entier $\ge 1$. Pour un nombre premier $p$ on note ${\mathbf{Q}}_p^{\mathrm{nr}}$ l’extension maximale non ramifiée de ${\mathbf{Q}}_p$. Supposons que $p^v$ divise exactement $N$. Alors, en utilisant les travaux de Carayol et la théorie du corps de classes local, on détermine une extension $E_v$ de ${\mathbf{Q}}_p^{\mathrm{nr}}$ sur laquelle la jacobienne $J_0$ de la courbe modulaire de $X_0\left(N\right)$ admet une réduction semi-stable, puis on donne une estimation de son degré.

We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.

We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.

We prove a new lower bound for the height of points on a subvariety $V$ of a multiplicative torus, which lie outside the union of torsion subvarieties of $V$. Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an $\epsilon$ the sharpest conjectures that can be formulated....