On the Mumford–Tate conjecture for 1-motives

Abstract

We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group.

On the second Tate–Shafarevich group of a 1-motive

“…The following theorem is a special case of Theorem 6.2 of [Jos11]. In the case G is an abelian variety it goes back to a result of Ribet ([Rib76], see Appendix 2 of [Hin88]).…”

“…This is then in general not a commutative, but just a nilpotent Lie algebra. The generalisation of Theorem 3.5 is Theorem 6.2 in [Jos11]. The subgroup D of G(k) has to be replaced by the group of so-called deficient points (loc.cit.…”

On unipotent radicals of motivic Galois groups

“…A result of Deligne (written by Jossen in the appendix of [33]) describes the subobject u(M ) of W −1 End(M ) as follows 6 . From now on, if there is no ambiguity, we shall simply write Hom (resp.…”

“…As stated above, Deligne (see [33,Appendix]) describes u(M ) in terms of extensions that arise naturally from the weight filtration on M : For each integer p, let E p (M ) be the p-th extension class of M given by Eq. ( 1), considered as an extension of the unit object ½ by Hom(M/W p M, W p M ).…”

“…The second added ingredient is that we use the fundamental theorem of Tannakian categories with ω •Gr W as the fiber functor (rather than using ω itself). Notice the difference in the nature of this type of argument and Deligne's argument in [33,Appendix], which explicitly constructs an extension of ½ by u(M ) that pushes forward to the total class of M . We should point out that the idea of working with the associated graded fiber functor already appears in [20], and since then has featured frequently in the literature, especially in the setting of categories of mixed Tate motives (e.g.…”