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Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?

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up vote 11 down vote accepted

This is an immediate consequence of Theorem 2.11 and Proposition 2.9 in Deligne's 'Hodge cycles on abelian varieties' (notes by Milne available here: http://www.jmilne.org/math/Books/DMOS.pdf

2.11 shows that every Hodge cycle on an abelian variety $A$ over a field $k$ embeddable in $\mathbb{C}$ is absolutely Hodge, and 2.9 shows that all absolutely Hodge cycles are defined over a finite extension of $k$. So it follows that all Hodge cycles on an abelian variety have canonical $l$-adic realizations that are defined over the same finite extension of $k$. In particular, an open sub-group of the Galois group fixes all Hodge cycles; that is, it maps into the Mumford-Tate group of $A$.

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thanks Keerthi! –  Adam Harris Dec 1 '11 at 20:17
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