Martin's Blog

Motivic Galois groups and periods

Posted by Martin Orr on Tuesday, 03 November 2015 at 16:00

In my last post, I discussed how the existence of a polarisation implies an upper bound for the transcendence degree of the extended period matrix of an abelian variety, namely the dimension of the general symplectic group \operatorname{GSp}_{2g} (where g is the dimension of the abelian variety). In this post, I will discuss how this can be generalised to take into account all algebraic cycles on the abelian variety. The group \operatorname{GSp}_{2g} is replaced by the motivic Galois group of the abelian variety, which I will define. I will also mention how Deligne's theorem on absolute Hodge cycles allows us to replace the motivic Galois group by the Mumford-Tate group.

Tate twists in Deligne's Principle A

Before defining the motivic Galois group, we need some technical details about Tate twists and how they feature in Deligne's Principle A.

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

Recall that Deligne's Principle A says that the Mumford-Tate group MT(A_\mathbb{C}) is equal to the subgroup of \operatorname{GL}(H_1(A(\mathbb{C}), \mathbb{Q})) consisting of elements for which all the Hodge classes in all the cohomology groups H^{2p}(A^r(\mathbb{C}), \mathbb{Q}) are eigenvectors. In the case of the Mumford-Tate group, knowing that a Hodge class is an eigenvector is sufficient to determine what its eigenvalue must be -- the associated class in the Tate twist H^{2p}(A^r(\mathbb{C}), \mathbb{Q})(p) has eigenvalue 1 for all elements of the Mumford-Tate group.

On the other hand, in Principle A for the \ell-adic algebraic monodromy group G_\ell(A), it is not enough just to say that Tate classes are eigenvectors for elements of G_\ell(A). We have to specify that they have eigenvalue 1:

Theorem. G_\ell(A) is the subgroup of \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) consisting of those elements which fix all Tate classes in H^{2p}_{et}(A^r, \mathbb{Q}_\ell)(p) for all positive integers p and r.

But this raises a question: how does \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) actually act on H^{2p}_{et}(A^r, \mathbb{Q}_\ell)(p) (and a similar question for singular cohomology)? The action on H^{2p}_{et}(A^r, \mathbb{Q}_\ell) is clear, because  H^{2p}_{et}(A^r, \mathbb{Q}_\ell) = \bigwedge^{2p} (T_\ell A \otimes \mathbb{Q}_\ell)^{\vee r}.

But how does it act on the Tate twist object \mathbb{Q}_\ell(p)? (The definition of \mathbb{Q}_\ell(p) tells us how \operatorname{Gal}(\bar{k}/k) acts, but not the more abstract algebraic group \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell).)

This is where we use a detail which I swept under the carpet before. Namely, in the theorem, we characterise G_\ell(A) as a subgroup of \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) (relative to a symplectic pairing \psi_\ell induced by a polarisation) rather than as a subgroup of \operatorname{GL}(T_\ell A \otimes \mathbb{Q}_\ell). Abstractly, there is a character  \chi \colon \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) \to \mathbb{G}_m telling us how \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) acts on the pairing \psi_\ell. We define the action of \operatorname{GSp}(T_\ell A \otimes \mathbb{Q}_\ell) on \mathbb{Q}_\ell(p) to be via the character \chi^p -- motivated by the fact that this choice of action matches the action on self-cup products of a polarisation.

The theorem works when we do this, but it is a little inelegant because it depends on choosing a polarisation of A. Deligne used a more elegant approach: we can characterise G_\ell(A) as a subgroup of the product  \operatorname{GL}(T_\ell A \otimes \mathbb{Q}_\ell) \times \mathbb{G}_m. We let \operatorname{GL}(T_\ell A \otimes \mathbb{Q}_\ell) act on the cohomology groups in the obvious way, and trivially on \mathbb{Q}_\ell(p), while the \mathbb{G}_m factor acts trivially on the cohomology groups and via the character z \mapsto z^p on the Tate twist object \mathbb{Q}_\ell(p).

Then Principle A for the \ell-adic algebraic monogromy group becomes:

Theorem. Let G denote the subgroup of \operatorname{GL}(T_\ell A \otimes \mathbb{Q}_\ell) \times \mathbb{G}_m consisting of those elements which fix all Tate classes in H^{2p}_{et}(A^r, \mathbb{Q}_\ell)(p) for all positive integers p and r. Then the projection G onto the first factor \operatorname{GL}(T_\ell A \otimes \mathbb{Q}_\ell) is injective and has image equal to G_\ell(A).

Of course we could make a similar statement for the Mumford-Tate group.

The motivic Galois group

We saw last time that the existence of a polarisation implies that the transcendence degree of the extended period matrix of an abelian variety A/k is bounded above by the dimension of the algebraic group \operatorname{GSp}_{2g}, that is, the group which preserves the symplectic form on H^1(A(\mathbb{C}), \mathbb{Q}). We can generalise this by introducing the motivic Galois group G^{mot}(A_\mathbb{C}).

Deligne's Principle A gave us theorems describing the Mumford-Tate group and the \ell-adic algebraic monodromy group as groups which fix Hodge classes and Tate classes respectively. For the motivic Galois group, we turn this around, and define G^{mot}(A_\mathbb{C}) to be the group which fixes algebraic cycle classes.

Specifically, G^{mot}(A_\mathbb{C}) is defined to be the subgroup of \operatorname{GL}(H_1(A(\mathbb{C}), \mathbb{Q})) \times \mathbb{G}_m which fixes the algebraic cycle classes  cl_B(Z) \in H^{2p}(A^r(\mathbb{C}), \mathbb{Q})(p) for all algebraic cycles Z on A_\mathbb{C} for all integers r and p. (The action of \operatorname{GL}(H_1(A(\mathbb{C}), \mathbb{Q})) \times \mathbb{G}_m is defined as in the previous section, with the \mathbb{G}_m there to take care of Tate twists.) For each algebraic cycle Z, this gives polynomial equations with coefficients in \mathbb{Q} which must be satisfied by elements of G^{mot}(A_\mathbb{C}), and so G^{mot}(A_\mathbb{C}) is an algebraic group over \mathbb{Q}.

Comparing the definition of the motivic Galois group with Principle A for the Mumford-Tate group, we observe that all algebraic cycle classes are Hodge classes and hence  MT(A_\mathbb{C}) \subset G^{mot}(A_\mathbb{C}). The Hodge conjecture is equivalent to MT(A_\mathbb{C}) = G^{mot}(A_\mathbb{C}). Similarly, the fact that all algebraic cycle classes are Tate classes implies that the \ell-adic algebraic monodromy group of A is contained in G^{mot}(A_\mathbb{C}) \times_{\mathbb{Q}} \mathbb{Q}_\ell, and the the Tate conjecture is equivalent to equality of these groups.

The fact that a polarisation is an algebraic cycle class implies that  G^{mot}(A_\mathbb{C}) \subset \operatorname{GSp}(H_1(A(\mathbb{C}), \mathbb{Q})) (embedded in \operatorname{GL}(H_1(A(\mathbb{C}), \mathbb{Q})) \times \mathbb{G}_m by the map g \mapsto (g, \chi(g)^{-1}) where \chi is the tautological character of \operatorname{GSp}(H_1(A(\mathbb{C}), \mathbb{Q}))).

Upper bound for the transcendence degree of periods

To obtain a bound for the transcendence degree of periods using the motivic Galois group, let us we introduce the motivic torsor of periods. We define \Omega_A^{mot}(K) (for each field K containing k) to be the set of isomorphisms of K-vector spaces  \alpha \colon H^1_{dR}(A/k) \otimes_k K \to H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} K with the property that the induced maps on H^{2p}(A^r_{\mathbb{C}})(p) (for all p and r) commute with the cycle class maps i.e.  \alpha(cl_{dR}(Z)) = cl_B(Z) \text{ in } H^{2p}(A^r(\mathbb{C}), \mathbb{Q})(p) \otimes_{\mathbb{Q}} \mathbb{C} for all algebraic cycles Z of codimension p on A_{\mathbb{C}}^r.

Because every de Rham cycle class cl_{dR}(Z) on A_{\mathbb{C}}^r is in fact defined over \bar{k}, while all the Betti cycle classes cl_B(Z) are defined over \mathbb{Q} which is contained in \bar{k}, it follows that \Omega_A^{mot} is the functor of points of a \bar{k}-scheme.

The \bar{k}-scheme \Omega_A^{mot} is a torsor for G^{mot}(A_\mathbb{C}) \times_\mathbb{Q} \bar{k}, that is, G^{mot}(A_\mathbb{C}) acts on \Omega_A^{mot} by composition of linear maps  H^1_{dR}(A/\bar{k}) \to^\alpha H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} \bar{k} \to^g H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} \bar{k} and this action is faithful and transitive (on \bar{k}-points). Hence  \dim \Omega_A^{mot} = \dim G^{mot}(A_\mathbb{C}).

Observe that the comparison isomorphism \sigma^* \colon H^1_{dR}(A/\mathbb{C}) \to H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} \mathbb{C} commutes with the cycle class maps and hence is a \mathbb{C}-point of \Omega_A^{mot}. With respect to a suitable coordinate system, its coordinates are given by the extended period matrix.

It is a standard result of algebraic geometry that the transcendence degree of a point in a variety is at most the dimension of the variety. Hence we have proved that  \operatorname{trdeg}(\text{extended period matrix of } A) \leq \dim G^{mot}(A_\mathbb{C}). As we have observed that G^{mot}(A_\mathbb{C}) \subset \operatorname{GSp}(H_1(A(\mathbb{C}), \mathbb{Q})), this implies the result mentioned last time that  \operatorname{trdeg}(\text{extended period matrix of } A) \leq \dim \operatorname{GSp}_{2g}.

The Grothendieck period conjecture is the conjecture that in fact  \operatorname{trdeg}(\text{extended period matrix of } A) = \dim G^{mot}(A_\mathbb{C}). Proving lower bounds for transcendence degrees is much harder than proving upper bounds, and indeed no non-trivial lower bounds are known except for elliptic curves.

Transcendence of periods and absolute Hodge classes

As remarked above, the Hodge conjecture implies that G^{mot}(A_\mathbb{C}) = MT(A_\mathbb{C}). Hence the Hodge conjecture implies that  \operatorname{trdeg}(\text{extended period matrix of } A) \leq \dim MT(A_\mathbb{C}). This is useful because \dim MT(A_\mathbb{C}) is generally easier to calculate than G^{mot}(A_\mathbb{C}).

Using absolute Hodge classes, we can prove the above inequality involving the Mumford-Tate group, even without knowing the Hodge conjecture. To do this, we introduce the absolute Hodge torsor of periods \Omega^{AH}_A. For every field K containing k, the K-points of \Omega^{AH}_A are defined to be the isomorphisms of K-vector spacese above inequality for abelian varieties.  \alpha \colon H^1_{dR}(A/k) \otimes_k K \to H^1(A(\mathbb{C}), \mathbb{Q}) \otimes_{\mathbb{Q}} K with the property that  \alpha(v) = \sigma^*(v) for every absolute Hodge class v \in H^{2p}_{dR}(A^r/\mathbb{C})(p), for all r and p.

The key points we need now are that:

  1. Absolute Hodge classes are defined over \bar{k}, so the variety \Omega^{AH}_A is defined over \bar{k}.
  2. By Deligne's theorem, Hodge classes on A are absolute Hodge classes, so the Mumford-Tate group acts faithfully and transitively on \Omega^{AH}_A.

Then the same arguments as before tell us that  \operatorname{trdeg}(\text{extended period matrix of } A) \leq \dim \Omega^{AH}_A = \dim MT(A_\mathbb{C}).

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

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