Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.)

share|improve this question
    
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. –  jmc Oct 11 '13 at 14:52
add comment

2 Answers

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.

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.