In this paper we study the structure and the degeneracies of the Mumford-Tate group $MT\left(M\right)$ of a 1-motive $M$ defined over $ℂ$. This group is an algebraic $\mathbb{Q}$- group acting on the Hodge realization of $M$ and endowed with an increasing filtration $W_{•}$. We prove that the unipotent radical of $MT\left(M\right)$, which is $W_{-1}\left(MT\left(M\right)\right)$, injects into a “generalized” Heisenberg group. We then explain how to reduce to the study of the Mumford-Tate group of a direct sum of 1-motives whose torus’character group and whose lattice are both of rank 1....

On donne une nouvelle démonstration directe du théorème de Hilbert-Samuel arithmétique et on déduit un critère numérique pour l’existence de sections d’un fibré en droite sur une variété arithmétique de norme sup inférieure à un.