Morphisms of 1-motives Defined by Line Bundles

Cristiana Bertolin (1) , Sylvain Brochard (2)

1 Laboratoire de Theorie de Nombres
2 IMAG - Institut Montpelliérain Alexander Grothendieck

Let 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.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Cristiana Bertolin, Sylvain Brochard. Morphisms of 1-motives Defined by Line Bundles. International Mathematics Research Notices, 2018, International Mathematics Research Notices, 2019 (Issue 5), pp.1568-1600. ⟨10.1093/imrn/rny139⟩. ⟨hal-01819644v2⟩