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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.

Tate twists in \ell-adic cohomology

In singular cohomology, using Tate twists is largely a matter of convenience as you can avoid them by putting in factors of 2 \pi i. On the other hand, using Tate twists in \ell-adic cohomology (for example, when defining the algebraic cycle class map) is essential. There are two reasons for this:

  1. The Tate twist object \mathbb{Z}(1) in the category of \mathbb{Z}-Hodge structures has a canonical generator 2 \pi i and so we get canonical isomorphisms of the underlying lattices V_\mathbb{Z} \cong V_\mathbb{Z}(m) for any Hodge structure V (at least, canonical up to sign). On the other hand, the Tate twist object \mathbb{Z}_\ell(1) in the \ell-adic world is a rank 1 \mathbb{Z}_\ell-module but has no canonical isomorphism to \mathbb{Z}_\ell, so V_\ell and V_\ell(m) are only non-canonically isomorphic.

  2. If we are working over a non-algebraically closed base field k, the Tate twist object has a non-trivial action of \operatorname{Gal}(\bar{k}/k) so V_\ell and V_\ell(m) are not isomorphic as Galois representations.

The \ell-adic Tate twist object \mathbb{Z}_\ell(1) is defined to be the inverse limit \varprojlim \mu_{\ell^n}(\bar{k}) where \mu_{\ell^n}(\bar{k}) is the group of \ell^n-th roots of unity in \bar{k}. Thus \mathbb{Z}_\ell(1) is a rank 1 \mathbb{Z}_\ell-module and \operatorname{Gal}(\bar{k}/k) acts on \mathbb{Z}_\ell(1) as multiplication by the cyclotomic character \chi_\ell (this is essentially the definition of \chi_\ell).

We define notations \mathbb{Z}_\ell(m), \mathbb{Q}_\ell(m) and V_\ell(m) for any \ell-adic Galois representation V_\ell in the same way as we defined analogous notations for the Tate twist of Hodge structures.

The \ell-adic cycle class map cl_\ell has image in H^{2p}_{et}(X_{\bar{k}}, \mathbb{Z}_\ell)(p) . The Tate twist is necessary here to get a Galois-equivariant cycle class map, because it has to map subvarieties defined over k to a \operatorname{Gal}(\bar{k}/k)-invariant classes, but there are no Galois-invariant classes in the untwisted H^{2p}_{et}(X_{\bar{k}}, \mathbb{Z}_\ell).

Over \mathbb{C}, there is a standard isomorphism \mathbb{Z}(1) \otimes_\mathbb{Z} \mathbb{Z}_\ell \to \mathbb{Z}_\ell(1) between the Betti and \ell-adic Tate twist objects, which is defined as the inverse limit of the system of maps z \mapsto \exp(z/\ell^n) : 2 \pi i \Z \to \mu_{\ell^n}(\mathbb{C}).

Hence if \sigma \colon \bar{k} \to \mathbb{C} is an embedding, we get comparison isomorphisms \sigma^* \colon H^{2p}_{et}(X_{\bar{k}}, \mathbb{Z}_\ell)(m) \to H^{2p}(X^\sigma, \mathbb{Z})(m) \otimes_{\mathbb{Z}} \mathbb{Z}_\ell.

Definition of absolute Hodge classes in \ell-adic cohomology

Because there is a comparison isomorphism between \ell-adic cohomology and singular cohomology for complex projective varieties, we can define absolute Hodge classes in \ell-adic cohomology with a definition which looks exactly the same as the definition for de Rham cohomology.

Let k be an algebraically closed field which can be embedded in \mathbb{C} and X a smooth projective variety over k. Deligne also defines absolute Hodge classes over non-algebraically closed base fields, but this definition does not seem particularly useful to me.

For each embedding \sigma \colon \bar{k} \hookrightarrow \mathbb{C}, we say that an \ell-adic cohomology class v \in H^{2p}_{et}(X, \mathbb{Z}_\ell)(p) is a Hodge class relative to \sigma if \sigma^*(v) \in H^{p,p}(X^\sigma)(p) \cap H^{2p}(X^\sigma, \mathbb{Z})(p), where \sigma^* is the comparison isomorphism between \ell-adic étale cohomology and singular cohomology.

We say that v \in H^{2p}_{et}(X, \mathbb{Z}_\ell)(p) is an absolute Hodge class in \ell-adic cohomology if v is a Hodge class relative to every embedding \sigma \colon k \hookrightarrow \mathbb{C}.

Deligne's theorem on absolute Hodge classes in \ell-adic cohomology

One can and prove state Deligne's theorem on absolute Hodge classes for \ell-adic cohomology just as for de Rham cohomology.

Theorem. Let k be an algebraically closed field k embeddable in \mathbb{C}. If A is an abelian variety over k and v \in H^{2p}_{et}(A, \mathbb{Z}_\ell)(p) is a Hodge class relative to one embedding \sigma \colon k \hookrightarrow \mathbb{C}, then v is an absolute Hodge class.

The proof uses Principle B for \ell-adic cohomology.

Theorem (Principle B). Let \pi \colon \mathcal{A} \to S be a family of abelian varieties over \mathbb{C}, with connected base S. Let v be a global section of R^{2p} \pi_* \mathbb{Q}(p).

If there is a point s \in S(\mathbb{C}) such that v_{s,et} is an absolute Hodge class on \mathcal{A}_s, then for every t \in S(\mathbb{C}), v_{t,et} is an absolute Hodge class on \mathcal{A}_t.

Here the notation v_{s,et} means that we evaluate the section v of R^{2p} \pi_* \mathbb{Q}(p) at s to get v_s \in H^{2p}(\mathcal{A}_s, \mathbb{Q})(p), then take the preimage of v_s under the comparison isomorphism H^{2p}_{et}(\mathcal{A}_s, \mathbb{Q}_\ell)(p) \to H^{2p}(\mathcal{A}_s, \mathbb{Q})(p) \otimes_{\mathbb{Q}} \mathbb{Q}_\ell.

The proofs we sketched before for Principle B and for Deligne's theorem work for \ell-adic cohomology with essentially no changes. This is because we used Blasius' statement and proof of Principle B, using the theorem of the fixed part. In Deligne's original paper, he had to use separate methods to prove Principle B for de Rham and for \ell-adic cohomology.

Application to the Mumford-Tate conjecture

Let A be an abelian variety over a number field k.

Recall that the Mumford-Tate conjecture is equivalent to the claim that potentially Tate classes are precisely the \mathbb{Q}_\ell-span of Hodge classes. We can use Deligne's theorem to prove one half of this, namely that all Hodge classes are potentially Tate.

In order to deduce this from Deligne's theorem, we just have to prove:

Lemma. Every absolute Hodge class on A_{\bar{k}} is a potentially Tate class on A.

The key point in the proof of the lemma is that the Galois action on H^{2p}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) preserves the absolute Hodge classes. This is simply because absolute Hodge classes are defined as classes satisfying a certain property for all embeddings \bar{k} \hookrightarrow \mathbb{C}, and \operatorname{Gal}(\bar{k}/k) permutes these embeddings.

Hence the action of \operatorname{Gal}(\bar{k}/k) on H^{2p}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) restricts to an action on the absolute Hodge classes. But the set of absolute Hodge classes is a finite-dimensional \mathbb{Q}-vector space because it is contained in H^{2p}(A^\sigma, \mathbb{Q})(p) for some \sigma. Thus we get a representation \operatorname{Gal}(\bar{k}/k) \to \operatorname{GL}_d(\mathbb{Q}). Furthermore, this representation is continuous for the \ell-adic topology on \operatorname{GL}_d(\mathbb{Q}) and its image is therefore a profinite group.

But any countable profinite group is finite. We deduce that the action of \operatorname{Gal}(\bar{k}/k) on absolute Hodge classes factors through a finite quotient, and hence that all absolute Hodge classes are potentially Tate.

Using Deligne's Principle A, we conclude that

Theorem. For any abelian variety A over a number field, the identity component of the \ell-adic algebraic monodromy group of A is contained in the extension of scalars of the Mumford-Tate group MT(A) \times_{\mathbb{Q}} \mathbb{Q}_\ell.

Tags abelian-varieties, alg-geom, hodge, maths, number-theory

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