# 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 (where 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 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 be an abelian variety defined over a number field .

Recall that Deligne's Principle A says that the Mumford-Tate group is equal to the subgroup of consisting of elements for which all the Hodge classes in all the cohomology groups 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 has eigenvalue 1 for all elements of the Mumford-Tate group.

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

Theorem. is the subgroup of consisting of those elements which fix all Tate classes in for all positive integers and .

But this raises a question: how does actually act on (and a similar question for singular cohomology)? The action on is clear, because

But how does it act on the Tate twist object ? (The definition of tells us how acts, but not the more abstract algebraic group .)

This is where we use a detail which I swept under the carpet before. Namely, in the theorem, we characterise as a subgroup of (relative to a symplectic pairing induced by a polarisation) rather than as a subgroup of . Abstractly, there is a character telling us how acts on the pairing . We define the action of on to be via the character -- 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 . Deligne used a more elegant approach: we can characterise as a subgroup of the product We let act on the cohomology groups in the obvious way, and trivially on , while the factor acts trivially on the cohomology groups and via the character on the Tate twist object .

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

Theorem. Let denote the subgroup of consisting of those elements which fix all Tate classes in for all positive integers and . Then the projection onto the first factor is injective and has image equal to .

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 is bounded above by the dimension of the algebraic group , that is, the group which preserves the symplectic form on . We can generalise this by introducing the motivic Galois group .

Deligne's Principle A gave us theorems describing the Mumford-Tate group and the -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 to be the group which fixes algebraic cycle classes.

Specifically, is defined to be the subgroup of which fixes the algebraic cycle classes for all algebraic cycles on for all integers and . (The action of is defined as in the previous section, with the there to take care of Tate twists.) For each algebraic cycle , this gives polynomial equations with coefficients in which must be satisfied by elements of , and so is an algebraic group over .

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 The Hodge conjecture is equivalent to . Similarly, the fact that all algebraic cycle classes are Tate classes implies that the -adic algebraic monodromy group of is contained in , and the the Tate conjecture is equivalent to equality of these groups.

The fact that a polarisation is an algebraic cycle class implies that (embedded in by the map where is the tautological character of ).

### 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 (for each field containing ) to be the set of isomorphisms of -vector spaces with the property that the induced maps on (for all and ) commute with the cycle class maps i.e. for all algebraic cycles of codimension on .

Because every de Rham cycle class on is in fact defined over , while all the Betti cycle classes are defined over which is contained in , it follows that is the functor of points of a -scheme.

The -scheme is a torsor for , that is, acts on by composition of linear maps and this action is faithful and transitive (on -points). Hence

Observe that the comparison isomorphism commutes with the cycle class maps and hence is a -point of . 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 As we have observed that , this implies the result mentioned last time that

The Grothendieck period conjecture is the conjecture that in fact 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 . Hence the Hodge conjecture implies that This is useful because is generally easier to calculate than .

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 . For every field containing , the -points of are defined to be the isomorphisms of -vector spacese above inequality for abelian varieties. with the property that for every absolute Hodge class , for all and .

The key points we need now are that:

1. Absolute Hodge classes are defined over , so the variety is defined over .
2. By Deligne's theorem, Hodge classes on are absolute Hodge classes, so the Mumford-Tate group acts faithfully and transitively on .

Then the same arguments as before tell us that