Maths > Abelian varieties > Periods of abelian varieties

## 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:

- Absolute Hodge classes are defined over
`, so the variety`

`is defined over`

`.`

- 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