Morphisms of 1-motives Defined by Line Bundles

International audienceLet S be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over S. We first compute a dévissage of the Picard group of a 1-motive M according to the weight filtration of M. This dévissage allows us to associate, to each line bundle L on M , a linear morphism ϕ L : M → M * from M to its Cartier dual. This yields a group homomorphism Φ : Pic(M)/Pic(S) → Hom(M, M *). We also prove the Theorem of the Cube for 1-motives, which furnishes another construction of the group homomorphism Φ : Pic(M)/Pic(S) → Hom(M, M *). Finally we prove that these two independent constructions of linear morphisms M → M * using line bundles on M coincide. However, the first construction, involving the dévissage of Pic(M), is more explicit and geometric and it furnishes the motivic origin of some linear morphisms between 1-motives. The second construction, involving the Theorem of the Cube, is more abstract but also more enlightening