Talk:Mumford–Tate group

Some references and additions

I don't really know anything about the topic, and I'm a bit crunched for time so I thought I'd make a few comments on the talk page. Firstly, a good reference for this article seems to be Deligne's first article in the DMOS book (available on Milne's website [1]). He defines Mumford–Tate groups in section 3. In section 5, he defines the Mumford–Tate group of an abelian variety as that associated to its Betti homology with Q coefficients. He also makes a comment related to how the Mumford–Tate group is related to the motivic Galois group: the Mumford–Tate group of A is the automorphism group of the fibre functor of the Tannakian subcategory (of the neutral Tannakian category of rational Hodge structures) generated by H₁(A,Q) and the Tate motive Q(1). Serre's article in Motives I gives a similar definition for the Mumford–Tate group of a motive. This is what I've come up with so far. RobHar (talk) 16:46, 29 April 2010 (UTC)

There a a few approaches, I think. The Tannakian category approach is sort of the ultimate: it does the Zariski closure for you, but the technology is not really required just to get MT defined. Phillip Griffiths defines "Hodge tensor", I suppose in an expect way via (p, p) type tensors in the various tensor powers; and then MT is linear automorphisms that preserve all Hodge tensors. A good way to talk to physicists. Of course Deligne's approach has the advantage that he gets some results (via absolute Hodge cycles, my hazy memories of 1979 suggest). In any case more technical details can be added as needed, to explain additional material. Charles Matthews (talk) 17:07, 29 April 2010 (UTC)