The Mumford-Tate group of 1-motives

In this paper we study the structure and the degeneracies of the Mumford-Tate group MT(M) of a 1-motive M defined over ℂ. This group is an algebraic ℚ-group acting on the Hodge realization of M and endowed with an increasing filtration W•. We prove that the unipotent radical of MT(M), which is W-1(MT(M)), 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. Next we classify and we study the degeneracies of MT(M), i.e., those phenomena which imply the decrement of the dimension of MT(M).