Galois theory for motives of niveau ≤ 1

Let T be a Tannakian category over a field k of characteristic 0 and π(T ) its fundamental group. In this paper we prove that there is a bijection between the ⊗-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T -schemes of π(T ). We apply this result to the Tannakian category T1(k) generated by motives of niveau ≤ 1 defined over k, whose fundamental group is called the motivic Galois group Gmot(T1(k)) of motives of niveau ≤ 1. We find four short exact sequences of affine group sub-T1(k)-schemes of Gmot(T1(k)), correlated one to each other by inclusions and projections. Moreover, given a 1-motive M , we compute explicitly the biggest Tannakian subcategory of the one generated by M , whose fundamental group is commutative. Mathematics Subject Classifications (2000). 14L15, 18A22