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.

Appendix: The Tannakian construction of P(M) (following P. Deligne)

I reproduce here in almost unaltered form a comment of P. Deligne, explaining our construction of the Lie algebra of the unipotent motivic fundamental group P(M) of a 1-motive M in terms of Tannakian formalism. I alone am to blame for mistakes.

The starting point is that in Definition 4.3 the point $\overline{u}$ should be seen as an extension of $W_{-1}\mathcal {E}\mathit {nd}(M)$ by the unit object ℤ. This extension can be constructed in a general setting as follows:

A.1

Let $\mathcal{T}$ be a Tannakian category in characteristic zero with unit object . Suppose each object of $\mathcal{T}$ has a functorial exhaustive filtration W compatible with tensor products and duals, the functor $\mathrm{gr}^{W}_{\ast}$ being exact. For any object M, the object $\mathcal {E}\mathit {nd}(M) = M^{\vee}\otimes M$ contains the subobject $W_{-1}\mathcal {E}\mathit {nd}(M)$. The filtration of $\mathcal {E}\mathit {nd}(M)$ is deduced from the filtration of M, hence

$$W_{-1}\mathcal {E}\mathit {nd}(M) = \mathrm{im} \Bigl(\bigoplus_p\mathcal {H}\mathit {om}(M/W_pM,W_pM) \to \mathcal {E}\mathit {nd}(M)\Bigr) $$

As , we can take the class of M in each of the vector spaces Ext¹(M/W _p M,W _p M), interpret it as a class in , take the sum of those and push to $W_{-1}\mathcal {E}\mathit {nd}(M)$ by functoriality of Ext¹. We get a class

which is natural in M under taking subobjects and quotients.

A.2

Let G be the fundamental group of $\mathcal{T}$. It has an invariant unipotent subgroup W ₋₁(G) with Lie algebra $W_{-1}(\operatorname{Lie} G)$. Thus, for all objects M, the group G acts on M respecting the filtration W, and the unipotent subgroup W ₋₁(M) acts trivially on $\mathrm{gr}^{W}_{\ast}M$. Let w:𝔾 _m →G/W ₋₁(G) be the cocharacter defined by the numbering of the filtration of W of M, for any M. This means that for a∈𝔾 _m , the automorphism w(a) of $\mathrm{gr}^{W}_{\ast}(M)$ acts as multiplication with a ⁿ on $\mathrm{gr}^{W}_{n}(M)$. Let us consider the inverse image of this 𝔾 _m in G, or more precisely $\mathbb {G}_{m}\times_{G/W_{-1}G}G$. Its Lie algebra $\mathfrak{L}$ is an extension of by $\operatorname{Lie} W_{-1}(G)$.

For any object M of $\mathcal{T}$, let $\mathfrak{u}_{M}$ be the image in $W_{-1}\mathcal {E}\mathit {nd}(M)$ of $\operatorname{Lie} W_{-1}(G)$. The extension $\mathfrak{L}$ gives us by functoriality a quotient $\mathfrak{L}_{M}$ of $\mathfrak{L}$ acting on M, which is an extension of by $\mathfrak{u}_{M}$.

In the setting of Sect. 4, U(M) corresponds to $W_{-1}\mathcal {E}\mathit {nd}(M)$, and the subobject P(M) of U(M) corresponds to the subobject $\mathfrak{u}_{M}$ of $W_{-1}\mathcal {E}\mathit {nd}(M)$. We have a 1-motive [ℤ→U(M)] given by the point $\overline{u}$, which has to be seen as an extension of [0→U(M)] by [ℤ→0], and P(M) was defined to be the smallest subobject of U(M) from which this extension comes, say after passing to ℚ-coefficients. Exactly in this way the two stories A.1 and A.2 are related:

Proposition A.3

• (1) The extension cl(M) is the push out of the extension $\mathfrak{L}_{M}$ via the inclusion $\mathfrak{u}_{M}\to W_{-1}\mathcal {E}\mathit {nd}(M)$.

• (2) If $\mathfrak{v}$ is a subobject of $\mathfrak{u}_{M}$ such that cl(M) is the image of a class in , then $\mathfrak{v} = \mathfrak{u}_{M}$.

I claim in (1) an equality in . As is 0 for weight reasons, it is the same as an (unique) isomorphism of actual extensions.

A first computation, which was helpful for understanding but now has disappeared from the proof, was to consider the category of graded representations of a graded Lie algebra $\mathfrak{g}$ with degrees <0, and wondering what cocycle $c:\mathfrak{g}\to W_{-1} \mathcal {E}\mathit {nd}(M)$ was giving me cl(M) in $H^{1}(\mathfrak{g}, W_{-1}\mathcal {E}\mathit {nd}(M))$. The answer is the composite map $\mathfrak{g}\to \mathfrak{g}\to W_{-1} \mathcal {E}\mathit {nd}(M)$, where the first map is multiplication by n in degree −n, and the second map is given by the action of $\mathfrak{g}$ on M.

Proof of Proposition A.3

(1) For each integer n we have a map

The push-out by $\mathfrak{u}_{M} \to W_{-1}\mathcal {E}\mathit {nd}(M)$ of the extension is the pull-back of the extension

$$ 1\to W_{-1}\mathcal {E}\mathit {nd}(M) \to W_0\mathcal {E}\mathit {nd}(M) \to \mathrm{gr}^W_0\!\mathcal {E}\mathit {nd}(M) \to 1 $$

(∗)

by ∑ _n ne _n . Indeed, ∑ _n ne _n gives the action of on $\mathrm{gr}^{W}_{\ast}\!M$ corresponding to the grading. Define E _p :=∑ _n>p e _n and let A≤B be integers such that W _A M=0 and W _B M=M. We have then

$$\sum_nne_n = \sum_{A\leq p\leq B} E_p + m\sum_ne_n $$

for some integer m depending on the choice of A and B. The map ∑ _n e _n lifts to the identity morphism of M, viewed as a map which factors over $W_{0}\mathcal {E}\mathit {nd}(M)$. The pull-back of (∗) by m∑ _n e _n is hence trivial, and the push-out of $\mathfrak{L}_{M}$ by the inclusion $\mathfrak{u}_{M} \to W_{-1}\mathcal {E}\mathit {nd}(M)$ is the sum of the pull-backs of (∗) by the E _p . This pull-back by E _p comes from an extension of by $\mathcal {H}\mathit {om}(M/W_{p}M,W_{p}M)$, which I would like to call Endomorphisms of M respecting W _p (M), inducing a multiple of the identity on M/W _p M and 0 on W _p M. At least, that is what it becomes in any realisation. This is the extension already considered in A.1, corresponding to the class of M in Ext¹(M/W _p M,W _p M). The sum of those extension classes, pushed to $W_{-1}\mathcal {E}\mathit {nd}(M)$, had been defined to be cl(M).

(2) On objects N of weights <0, i.e. such that N=W ₋₁ N, the functor is left exact. For any class α in , there is hence a smallest sub-object N ₀ of N such that α comes from a class in . Indeed, if α comes from N′ and from N″, the short exact sequence

$$0\to N'\cap N'' \to N'\oplus N'' \to N $$

shows after applying that α also comes from N′∩N″. It remains to show that the class of $\mathfrak{L}_{M}$ in does not come from any $\mathfrak{v} \subsetneq \mathfrak{u}_{M}$. Indeed, any subobject of $\mathfrak{L}_{M}$ is stable by the action of $\mathfrak{L}_{M}$ because G and $\operatorname{Lie} G$ act on everything. □

Proposition A.4

Any Lie-subobject of $\mathfrak{L}_{M}$ mapping onto is equal to $\mathfrak{L}_{M}$. In other words, the extension is essential.

Proof

The bracket of $\mathfrak{L}_{M}$ passes to the quotients by W and defines brackets

$$W_p \mathfrak{L}_M /W_{p-1} \mathfrak{L}_M \otimes \mathfrak{L}_M /W_{-1} \mathfrak{L}_M \to W_p \mathfrak{L}_M /W_{p-1}\mathfrak{L}_M $$

and this bracket is the multiplication by p. For any subobject $\mathfrak{L}'$ of $\mathfrak{L}_{M}$ mapping onto the quotient , the stability of $\mathfrak{L}'$ by the action of $W_{p} \mathfrak{L}_{M}$ hence gives a surjectivity

$$\mathrm{gr}^W_p \mathfrak{L}' \to \mathrm{gr}^W_p \mathfrak{L}_M $$

from which the equality $\mathfrak{L}' = \mathfrak{L}_{M}$ follows. □