Let $A/K$ be an abelian variety over a number field. On the one hand we have the singular cohomology group

$$V := H^1(A(\mathbb{C}),\mathbb{Q})$$

with respect to some fixed embedding $K \subset \mathbb{C}$, and from this we can construct the Mumford-Tate group $G_A \subset Aut_\mathbb{Q}(V)$, a reductive algebraic group over $\mathbb{Q}$, associated to the natural Hodge structure of $V$.

On the other hand, for each prime $l$, we have the $l$-adic étale cohomology group

$$V_l := H^1(A(\overline{K}),\mathbb{Q}_l)$$

with respect to some fixed algebraic closure $\overline{K}$ of $K$, and from this we can construct the $l$-adic algebraic monodromy group $G_{A,l} \subset Aut_{\mathbb{Q}_l}(V_l)$, a reductive algebraic group over $\mathbb{Q}_l$, defined as the Zariski closure of the image of the $G_K$-representation acting continuously on $V_l$ (this latter representation being dual to that on the Tate module). The comparison isomorphism $V_l \cong V \otimes_\mathbb{Q} \mathbb{Q}_l$ allows us to compare the identity component $G^0_{A,l}$ of $G_{A,l}$ with $G_{A,\mathbb{Q}_l}$ (that is, $G_A$, but viewed as an algebraic group over $\mathbb{Q}_l$), and the Mumford-Tate conjecture predicts that these two groups are the same.

Here is my question:

What are some consequences of the Mumford-Tate conjecture for the arithmetic of abelian varieties?

An example of the sort of thing I have in mind is the following recent result of David Zywina: if $A$ as above is absolutely simple, $K$ is large enough, and the Mumford-Tate conjecture for $A$ holds, then the density of good primes $v$ of $K$ where the reduction $A_v/\mathbb{F}_v$ is also absolutely simple is 1 if and only if the endomorphism ring of $A$ is commutative. (Zywina's result is actually finer than this, see the link.)

I guess I'm asking for other places in the literature where results of this flavour are proven to follow from MTC.

(A word on my motivation: I have a problem regarding abelian varieties which I think may follow from MTC, but I'm rather stuck proving this. I think my cause would be helped if I had more tools available.)

  • $\begingroup$ Excuse me, but how do we know that $G_{A,l}$ is reductive? As far as I know this is an open problem, because on the $l$-adic side we do not have a good analogue of polarisations. If you have a reference for this result, I would be very grateful. $\endgroup$
    – jmc
    Oct 11 '13 at 14:52

Here is a result in a somewhat different direction. Assuming MTC, Hindry and Ratazzi recently proved a bound for the torsion subgroup of abelian varieties of ${\rm GSp}$-type over arbitrary finite extensions.


does not Mumford-Tate conjecture mean that Hodge conjecture implies Tate conjecture? Did you have something more in mind?


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