Martin's Blog

Absolute Hodge classes in l-adic cohomology

Posted by Martin Orr on Friday, 26 June 2015 at 11:30

We can define absolute Hodge classes in \ell-adic cohomology in the same way as absolute Hodge classes in de Rham cohomology. We can then prove Deligne's theorem, that Hodge classes on an abelian variety are absolute Hodge, for \ell-adic cohomology. Because it is easy to prove that absolute Hodge classes in \ell-adic cohomology are potentially Tate classes, this implies half of the Mumford-Tate conjecture.

In particular, it implies that if A is an abelian variety over a number field, then a finite index subgroup of the image of the \ell-adic Galois representation on T_\ell A is contained in the \mathbb{Q}_\ell-points of the Mumford-Tate group of A. This is the goal I have been working towards for some time on this blog.

Deligne's definition of absolute Hodge classes considered \ell-adic cohomology (for all \ell) and de Rham cohomology simultaneously. The accounts I read of this theory focussed on the de Rham side, leading me to believe that the de Rham part was essential and the \ell-adic part an optional extra. This is why I wrote the past few posts about de Rham cohomology and am now adding the \ell-adic version on at the end, even though I am more interested in the \ell-adic version. Now that I understand what is going on, I realise that I could have used only \ell-adic cohomology from the beginning. One day I might write up a neater account which uses \ell-adic cohomology only.

Before the main part of this post, talking about absolute Hodge classes in \ell-adic cohomology, I need to talk about Tate twists in \ell-adic cohomology. These are more significant than Tate twists in singular cohomology because they change the Galois representations involved. This resulted in some mistakes in my previous posts on Tate classes, which I think I have now fixed.

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

Tate twists in singular and de Rham cohomology

Posted by Martin Orr on Friday, 19 June 2015 at 19:30

Tate twists in singular cohomology are a device for dealing with factors of 2 \pi i which come up whenever we compare singular and de Rham cohomology of complex projective varieties. In this post I will explain the problem, including calculating the 2 \pi i in the case of \mathbb{P}^1, and define Tate twists to solve it.

In the case of singular cohomology, Tate twists are largely a matter of normalising things conveniently. Without them, we could just write out factors of 2 \pi i everywhere. On the other hand, there is also a notion of Tate twists for \ell-adic cohomology, which cannot be omitted so easily, and which I will discuss in a subsequent post.

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

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