Martin's Blog

Images of Galois representations

Posted by martin 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.

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Tate modules

Posted by martin on Sunday, 21 November 2010 at 17:32

I said after my last post that I would write something about $\ell$-adic representations coming from abelian varieties. I have finally got around to doing so: here I will tell the story of how these representations are defined, and show that the Tate module is canonically isomorphic to $H_1(A, \mathbb{Z}) \otimes \mathbb{Z}_\ell$. Next time I will relate this to Mumford-Tate groups.

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