Martin's Blog

Deligne's Principle A and the Mumford-Tate conjecture

Posted by Martin Orr on Wednesday, 10 September 2014 at 11:20

In this post I will fill in a missing detail from two weeks ago, where I mentioned that the Mumford-Tate group is determined by the Hodge classes. More precisely, I will show that an element g of \operatorname{GL}(H_1(A, \mathbb{Z})) is in the Mumford-Tate group if and only if every Hodge class on every Cartesian power A^r is an eigenvector of g. In the context of Deligne's theorem on absolute Hodge classes, this is known as Principle A.

We will also see that a version of this statement holds for the \ell-adic monodromy group and Tate classes. This implies a link between the Hodge, Tate and Mumford-Tate conjectures.

no comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths, number-theory Read more...

Images of Galois representations

Posted by Martin Orr on Saturday, 27 November 2010 at 16:22

In this post, I will continue to talk about the \ell-adic representations attached to abelian varieties, and in particular the images G_\ell of these representations. I will define algebraic groups approximating G_\ell, which are often more convenient to work with. I will end by stating the Mumford-Tate conjecture, linking G_\ell to the Mumford-Tate group.

7 comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths Read more...

Mumford-Tate groups

Posted by Martin Orr on Monday, 04 October 2010 at 12:37

In this post I will define the Mumford-Tate group of an abelian variety. This is a \mathbb{Q}-algebraic group, such that the Hodge structure is a representation of this group. The Mumford-Tate group is important in the study of Hodge theory, and surprisingly also tells us things about the \ell-adic representations attached to the abelian variety.

4 comments Tags abelian-varieties, alg-geom, alg-groups, hodge, maths Read more...

Representations of algebraic tori

Posted by Martin Orr on Wednesday, 03 March 2010 at 10:08

I am finally ready to finish my series on algebraic tori, by talking about their representations. I shall show that these representations can be classified by a grading on the vector space of the representation, after extending scalars to the separable closure. I will describe this classification explicitly in a simple case.

no comments Tags alg-geom, alg-groups, maths, representations, tori Read more...

Character groups of algebraic tori

Posted by Martin Orr on Sunday, 24 January 2010 at 18:10

In this post I will return to the subject of algebraic tori. Just as Pontryagin duality classifies locally compact abelian groups through their characters, so algebraic tori are also classified by their characters.

In order to account for the arithmetic phenomenon of non-split tori, we need to include a Galois action on the character group. The primary result of this post is that there is an anti-equivalence of categories between {k-tori} and {finitely generated free abelian groups with a continuous action of \mathop{\mathrm{Gal}}(k^s/k)}.

2 comments Tags alg-geom, alg-groups, maths, tori Read more...

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