Martin's Blog

Tate modules

Posted by Martin Orr 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.

2 comments Tags abelian-varieties, alg-geom, maths, number-theory

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

Hodge structures and abelian varieties

Posted by Martin Orr on Friday, 24 September 2010 at 08:48

I spend most of my time thinking about the Hodge structures attached to abelian varieties, so I decided that I should explain what these Hodge structures are. A Hodge structure is a type of algebraic structure found on the (co)homology of complex projective varieties.

Here I will discuss only the special case of the first homology of abelian varieties. This is the simplest case, but is nonetheless very important. In particular, the Hodge structures on other homology and cohomology groups for abelian varieties can be calculated from that of the first homology. Also Hodge structures on the first (but not higher) cohomology of non-abelian varieties can be reduced to the case of abelian varieties by passing to something called the Albanese variety, generalising the Jacobian of curves.

2 comments Tags abelian-varieties, alg-geom, hodge, maths

M2 thesis and defence

Posted by Martin Orr on Friday, 13 August 2010 at 15:05

Last month I finished my Masters in Orsay. Since March I had been working on a thesis on a 1998 paper of Pink on the Mumford-Tate conjecture (concerning l-adic Galois representations attached to abelian varieties). The paper was difficult to read, but I learnt a lot doing so.

In July I defended the thesis in Orsay. This did not seem a valuable exercise. I spoke for a bit under an hour to my adviser, one other examiner and one of my friends (in my case the second examiner was my adviser's collaborator, visiting from UCL). In principle the public are permitted to attend, but since it is not announced anywhere noone is likely to do so except some friends you may have invited - I can't imagine anyone else wanting to attend anyway.

Since I had already explained all the contents of my thesis to my adviser in our weekly meetings, the only person I was really talking to was the second examiner. But as I understand it, the written thesis is the part which is most important for marking, and I doubt whether the second examiner reads that in detail. They did not ask many questions, and the questions they did ask I was mostly unable to answer - since I had focussed all my energies into understanding the proofs of certain specific results, and then writing down what I had understood, it would have been difficult to ask questions I could answer.

In September I will be returning to do a PhD with the same adviser, and working on the same sorts of questions.

no comments Tags abelian-varieties, m2, paris

Complex abelian varieties and Riemann forms

Posted by Martin Orr on Wednesday, 30 December 2009 at 21:48

The theory of abelian varieties is very beautiful, both in its arithmetic and geometrical aspects, and also looking just over \mathbb{C} where there are nice applications of complex analysis. In this post I will work over \mathbb{C}, and sketch a proof that a complex torus is isomorphic to an abelian variety if and only if it admits a Riemann form. This will assume some knowledge of the theory of complex manifolds.

1 comment Tags abelian-varieties, alg-geom, maths

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