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Tate classes

Posted by Martin Orr on Tuesday, 02 September 2014 at 19:30

In my last post I talked about Hodge classes on abelian varieties. Today I will talk about the analogue in \ell-adic cohomology, called Tate classes. Tate classes are defined to be classes in a Tate twist of the \ell-adic cohomology on which the absolute Galois group of the base field acts trivially.

The Tate classes on a variety change if we extend the base field (because this changes the Galois group). They are mainly interesting in the case in which the base field is finitely generated. In this post I will also define potentially Tate classes, which depend less strongly on the base field (they are unchanged by finite extensions).

I will state the Tate conjecture, the \ell-adic analogue of the Hodge conjecture, which says that if the base field is finitely generated, then the vector space of Tate classes is spanned by classes of algebraic cycles. I will also mention some other conjectures which are implied by or equivalent to the Tate conjecture or a slight strengthening of it.

Unlike in the case of Hodge classes, we cannot easily ignore the Tate twist in the definition of Tate classes. This post only contains brief remarks on Tate twists; there is a link to a later post with a more detailed discussion.

\ell-adic cohomology of abelian varieties

Let A be a smooth projective variety defined over a field k, and let \ell be a prime not equal to the characteristic of k. The machinery of étale cohomology gives us the \ell-adic cohomology groups H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell). Each of these groups comes with an action of \operatorname{Gal}(\bar{k}/k). Furthermore, if we have an embedding \bar{k} \hookrightarrow \mathbb{C}, then we get a comparison isomorphism with singular cohomology H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell) \cong H^n(A(\mathbb{C}), \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{Z}_\ell.

Now suppose that A is an abelian variety. In this case, we can easily compute the \ell-adic cohomology groups in terms of the \ell-adic Tate module (just as we can obtain the singular cohomology groups of a complex abelian variety from H_1(A, \mathbb{Z})). Specifically we have H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell) = \bigwedge^n (T_\ell A)^\vee and the representation of \operatorname{Gal}(\bar{k}/k) on T_\ell A induces the correct representation on H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell). Furthermore, if we have an embedding \bar{k} \hookrightarrow \mathbb{C}, then the comparison isomorphism between \ell-adic and singular cohomology is compatible with the natural isomorphism T_\ell A \cong H_1(A(\mathbb{C}), \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{Z}_\ell.

Similarly to the case of Hodge classes, it will be more convenient for us to work with cohomology groups with coefficients in \mathbb{Q}_\ell instead of \mathbb{Z}_\ell.

Tate twists

When defining Hodge classes as I did in the last post, it is possible to omit the use of Tate twists. If we try doing this for Tate classes we run into trouble as soon as we try to introduce cycle classes, and in particular cannot state the Tate conjecture. So I will give some very brief remarks on Tate twists here.

Later in this series, I have written a post on Tate twists for singular and de Rham cohomology, and a post which includes a fuller section on Tate twists for \ell-adic cohomology. I apologise for the forward references. This is because when I first wrote this post I did not realise the importance of Tate twists, so I wrote some incorrect things, and now I can't easily restructure the entire blog series.

Let \chi_\ell \colon \operatorname{Gal}(\bar{k}/k) \to \mathbb{Q}_\ell^\times denote the \ell-adic cyclotomic character i.e. \sigma \zeta = \zeta^{\chi_\ell(\sigma) \bmod \ell^n} for every \ell^n-th root of unity \zeta, for every n \in \mathbb{N} and \sigma \in \operatorname{Gal}(\bar{k}/k).

Given integers n and p, the Tate twist H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell)(p) is a \mathbb{Z}_\ell-module which is non-canonically isomorphic to H^n_{et}(A_{\bar{k}}, \mathbb{Z}_\ell) but with the action of \operatorname{Gal}(\bar{k}/k) twisted by the p-th power of the cyclotomic character.

Tate classes

We say that a class v \in H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) is a Tate class if it is fixed by the action of \operatorname{Gal}(\bar{k}/k).

This definition resembles the characterisation of Hodge classes as eigenvectors for the Mumford-Tate group in H^{2p}(A, \mathbb{Q}). However, unlike in the cases of Hodge classes, there may be eigenvectors for \operatorname{Gal}(\bar{k}/k) in H^{2p}(A_{\bar{k}}, \mathbb{Q}_\ell) with different characters, so it is necessary to specify that only fully \operatorname{Gal}(\bar{k}/k)-invariant classes are Tate classes.

Observe that the interest of this concept depends on the field k: if k is algebraically closed then all classes are Tate. In general we are interested in Tate classes when the field k is finitely generated of any characteristic (i.e. a number field, finite field or function field over one of these).

The Tate conjecture

One can define a cycle class map for the \ell-adic cohomology just like for the singular cohomology: for every subvariety Z \subset A defined over \bar{k} of codimension p, we get a class cl_\ell(Z) \in H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) . The cycle class map is compatible with Galois actions, so if Z is defined over k then cl_\ell(Z) is a Tate class. (Note that in order to get this compatibility with Galois actions, we have to twist H^{2p} by \chi_\ell^p - this is the reason why we need Tate twists.)

If we have an embedding \bar{k} \to \mathbb{C}, then the algebraic cycle classes in singular cohomology and in \ell-adic cohomology are compatible via the comparison isomorphism (if we take into account Tate twists correctly).

Tate Conjecture. If k is a finitely generated field, then the classes cl_\ell(Z) of algebraic subvarieties Z \subset A of codimension p defined over k span the \mathbb{Q}_\ell-vector space of Tate classes in H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p).

Of course this conjecture applies to all smooth projective varieties, not only abelian varieties.

The Tate conjecture looks similar to the Hodge conjecture. In the paper in which he first stated this conjecture (page 37 of these notes), Tate wrote:

I can see no direct logical connection between [the Tate conjecture] and Hodge’s conjecture that a rational cohomology class of type (p,p) is algebraic. ... However, the two conjectures have an air of compatibility.

The theory of absolute Hodge classes, which I will discuss in subsequent posts, supplies the missing direct logical connection between the Tate and Hodge conjectures.

Related conjectures

The Tate conjecture is related to several more concrete conjectures, which Tate discussed in his paper. There is also a nice explanation of these links in Milne's article on the work of Tate.

For example, if the base field k is a finite field, then the Tate conjecture, combined with an additional conjecture about the semisimplicity of the action of Frobenius on \ell-adic cohomology, is equivalent to a conjecture on the orders of the poles of the zeta function of the variety. (The semisimplicity conjecture is known for abelian varieties.)

Tate also discussed links with the Birch and Swinnerton-Dyer conjecture and the Sato-Tate conjecture.

The p=1 case of the Tate conjecture for abelian varieties is equivalent to the following theorem on endomorphisms. This statement is often referred to as the Tate conjecture (indeed I did so back here), but it is much weaker than the Tate conjecture stated above.

Theorem. \operatorname{End}_{\operatorname{Gal}(\bar{k}/k)} T_\ell A = \operatorname{End} A \otimes_\mathbb{Z} \mathbb{Z}_\ell.

This theorem on endomorphisms was proved by Tate over finite fields, which was the first case of the Tate conjecture to be proved. It was subsequently proved by Zarhin for finitely generated fields of positive characteristic and by Faltings for finitely generated fields of characteristic zero.

If we compare with the Hodge conjecture, we find that: The p=1 case of the Hodge conjecture is trivial for abelian varieties and known for varieties in general. The p=1 case of the Tate conjecture is a hard theorem for abelian varieties and still open for varieties in general. Thus the Tate conjecture appears to be harder than the Hodge conjecture.

Potentially Tate classes

I don't think I have ever seen the phrase "potentially Tate class" and Google doesn't find any hits, but it seems to me to describe something useful and fit the way "potentially" is often used by number theorists.

The issue is that, as we mentioned above, which classes are Tate depends on the base field. If A is defined over k and K is an extension of k, then A \times_k K may have more Tate classes than A (note that H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) itself is independent of the base field, it is just the Galois group appearing in the definition of Tate class which changes). This causes a problem when we try to compare Tate classes (for a variety defined over a number field say) with Hodge classes, which are about what happens over \mathbb{C}.

So we define a potentially Tate class on a smooth projective variety A defined over a field k to be a class v \in H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p) such that there is some finite extension K/k for which v is a Tate class on A \times_k K. (In other words, there is an open subgroup G = \operatorname{Gal}(\bar{k}/K) \subset \operatorname{Gal}(\bar{k}/k) which fixes v.) We only allow finite extensions because we know that if we go all the way to \bar{k} then all classes become Tate.

It is sometimes useful to know that, for any given variety A, there is a single finite extension K/k over which all the potentially Tate classes become Tate, not just on A itself but on every power of A. The proposition as I will state and prove it applies to each prime \ell separately, but in fact Serre showed that there is a single finite extension K/k which works for all \ell.

Proposition. Let A/k be a smooth projective variety. There exists a finite extension K/k such that every \ell-adic potentially Tate class on A^n (for every n \in \mathbb{N}) is a Tate class on A^n \times_k K.

Proof. If we only had to show this for A itself, without powers of A, then it would be easy: there are finitely many non-zero \ell-adic cohomology groups, and they are all finite dimensional, so we can choose a finite set of classes v_1, \dotsc, v_r which span all the potentially Tate classes on A. Then we just take a field k_i over which v_i becomes Tate for each i, and let K be the field generated by the k_i.

The above argument does not prove the proposition, because there are infinitely many cohomology groups of A^n as n varies. I shall describe an argument only for abelian varieties, although can be extended to cover all varieties. The convenience of working with abelian varieties is that we can build all the cohomology groups of all A^n out of H_1(A). In particular, H^{2p}_{et}(A_{\bar{k}}^n, \mathbb{Q}_\ell) \cong \bigwedge^{2p} \bigl( H_1(A_{\bar{k}}, \mathbb{Q}_\ell) \bigr)^{\vee \oplus n}. Hence any element of \operatorname{GSp}(H_1(A_{\bar{k}}, \mathbb{Q}_\ell)) naturally acts on all H^{2p}_{et}(A_{\bar{k}}^n, \mathbb{Q}_\ell)(p). (I am ignoring exactly how the Tate twist is dealt with here. It is due to this issue that it is only elements of \operatorname{GSp}(H_1(A)) which act, and not all of \operatorname{GL}(H_1(A)), where \operatorname{GSp} is defined as the general symplectic group with respect to some chosen Weil pairing.)

Define the \ell-adic algebraic monodromy group of A to be the Zariski closure (in \operatorname{GL}_{2g}(\mathbb{Q}_\ell)) of the image of the representation \rho_\ell \colon \operatorname{Gal}(\bar{k}/k) \to \operatorname{GSp}(H_1(A_{\bar{k}}, \mathbb{Q}_\ell)). Note that I often define the \ell-adic algebraic monodromy group in a similar way, but without the Tate twist; this makes only a small difference.

The \ell-adic algebraic monodromy group, denoted G_\ell, is an algebraic group and so has finitely many connected components. Let K be the finite extension of k such that \operatorname{Gal}(\bar{k}/K) = \rho_\ell^{-1}(G_\ell^\circ) where G_\ell^\circ is the identity connected component of G_\ell.

Let v \in H^{2p}_{et}(A_{\bar{k}}^n, \mathbb{Q}_\ell)(p) be a potentially Tate class, and let H_v = \{ g \in \operatorname{GSp}(H_1(A_{\bar{k}}, \mathbb{Q}_\ell)) : gv = v \}. The group H_v is Zariski closed because its defining condition is algebraic.

Because H_v is potentially Tate, \rho_\ell^{-1}(H_v) is a finite index subgroup of \operatorname{Gal}(\bar{k}/k). Because H_v is Zariski closed, it follows that H_v \cap G_\ell is a finite index subgroup of G_\ell, and hence contains G_\ell^\circ.

Thus \rho_\ell^{-1}(H_v) contains \rho_\ell^{-1}(G_\ell^\circ), i.e. v is a Tate class on A^n \times_k K.

We can restate the Tate conjecture in terms of potentially Tate classes as follows:

Conjecture. If k is a finitely generated field, then the classes cl_\ell(Z) of algebraic subvarieties Z \subset A_{\bar{k}} of codimension p defined over the algebraic closure \bar{k} span the \mathbb{Q}_\ell-vector space of potentially Tate classes in H^{2p}_{et}(A_{\bar{k}}, \mathbb{Q}_\ell)(p).

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

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