## Periods of abelian varieties

Posted by Martin Orr on Tuesday, 06 October 2015 at 16:10

There are a couple of different matrices associated with an abelian variety which are referred to as its period matrix.
These matrices relate different choices of bases for the tangent space or ```
of the abelian variety.
In this post I will discuss the different definitions of period matrices and how they relate to each other.
```

A complex abelian variety ` can be realised as the quotient of the -dimensional complex vector space `

` by the rank- lattice `

```
.
The period matrix expresses a basis of this lattice in terms of a basis for the tangent space.
We can also get a period matrix which is twice as large by using de Rham cohomology
```

` instead of the tangent space `

`.`

The period matrix can be defined for any complex abelian variety, but it contains additional information if the abelian variety is defined over a number field , as we can then choose our basis for ```
to also be defined over .
The period matrix then gives us a set of complex numbers relating a basis of a -vector space to a basis of a -vector space, and the transcendence properties of these numbers are interesting and I will discuss them in a later post.
```

### The period lattice

Let's work first over the complex numbers.
Let be a complex abelian variety of dimension .
Classically, a *period lattice* of is defined to be a lattice ` such that `

`.`

Recall that the lattice is the kernel of the exponential map ```
.
This is where the name
```

*period lattice* comes from: ` consists of the periods if `

```
in the sense that
```

```
In order to identify
```

` with a lattice in `

`, we have to choose a basis for `

```
.
Thus the period lattice is not uniquely defined, but only up to a change of basis for
```

`, that is, up to multiplication by an element of `

`.`

For example, the period lattice of an elliptic curve is a rank-2 lattice in , defined up to multiplication by an element of .

### Fundamental periods

Rather than an infinite object like a lattice, it can be useful to represent the same information in a finite list of numbers.
We can do this by choosing a basis for .
Then the coordinates of this basis (relative to our chosen basis for `) form a `

```
complex matrix.
The basis vectors are called
```

*fundamental periods* and the matrix of their coordinates is called a *period matrix* for .

It is often helpful to restrict our choice of basis for to only allow bases which are symplectic with respect to a chosen polarisation of .
Recall that a polarisation is a symplectic form
```
satisfying some conditions.
We can identify with
```

```
.
A
```

*symplectic basis* means a basis ` for `

```
such that
```

Once we have fixed the polarisation, there are two ambiguities in the definition of the period matrix of a polarised abelian variety: we can multiply by an element of ` on the left (changing the basis for `

`) and by an element of `

` on the right (changing the symplectic basis for `

`).`

### The Siegel matrix

The Siegel matrix of a complex abelian variety is obtained by choosing our basis for ` to be the same as the first half `

` of our symplectic basis for `

```
(it is a theorem that these vectors are -linearly independent).
Thus, the first columns of the period matrix simply form the identity matrix, and the
```

*Siegel matrix* is defined to be the second columns of the period matrix.

Thus the Siegel matrix is a complex matrix.
It is defined up to an action of ` (changing the symplectic basis for `

```
) given by
```

```
The Riemann bilinear relations assert that the Siegel matrix is symmetric and has positive definite imaginary part (i.e. it lives in the Siegel upper half-space
```

```
).
This leads to the construction of the moduli space of principally polarised abelian varieties as
```

`.`

Note: the Siegel matrix is sometimes called the Riemann matrix and that is perhaps more sensible, but the term Riemann matrix often refers to the period matrix above, so I have chosen the name of Siegel matrix to avoid ambiguity.

### Arithmetic information

Now let's suppose that our abelian variety is defined over a number field .
The tangent space ```
is a -vector space, and so we can choose a -basis for it and use that basis when computing the period matrix as above.
The lattice embeds into
```

`, not `

`, and so the period matrix still has complex entries (they are usually transcendental).`

Now it is defined up to multiplication by ` on the left and `

```
on the right.
Note that the field generated (over ) by the entries of the period matrix is not changed by these ambiguities, and in particular the transcendence degree of the periods is a well-defined invariant of which I will say more about in the next post.
```

### Calculating the periods

This abstract definition of periods as the kernel of ```
is all very well, but can we give a more concrete description?
We can do this using the fact that the "evaluation at 0" map
```

```
is an isomorphism when is an abelian variety (over any field of characteristic zero).
Here
```

` is the space of regular differential 1-forms on .`

The choice of a basis for ` induces a dual basis of `

`, and hence a basis `

` for `

```
.
To calculate the coordinates of a period
```

`, we simply have to evaluate the elements of the dual basis for `

` at `

```
.
Passing through the isomorphisms
```

` and `

```
, this means computing the integrals
```

For example, for an elliptic curve defined by an equation
```
the space of differential forms
```

` is generated by the regular differential form `

```
and we can calculate the fundamental periods as
```

### De Rham cohomology and the extended period matrix

There is another definition commonly used for the term period matrix.
This starts from the comparison isomorphism between de Rham cohomology and singular cohomology:
```
We can choose a -basis for
```

` and a -basis for `

`, then write down the matrix for the above isomorphism in terms of these bases.`

This gives a complex matrix, sometimes also called the period matrix.
In order to avoid confusion I shall call this one the *extended period matrix*.
It is defined up to multiplication by ` on the left and by `

```
on the right.
Once again the field generated over by the entries of the extended period matrix is independent of the choices of basis.
```

### Relationship between the two period matrices

How do the period matrix and extended period matrix relate to each other?
There is an injection
```
whose image is the component
```

```
of the Hodge filtration.
This follows from the fact that that the Hodge filtration on the de Rham of an abelian variety matches up with the inclusion
```

So we can take a basis for ` and extend it to a basis for `

```
.
We can also take a basis for
```

` and dualise to get a basis for `

```
, and use these bases to write down the extended period matrix.
Thus we see that, if we choose the bases appropriately, the period matrix defined using the period lattice is equal to the transpose of the left half of the extended period matrix.
(The "transpose" comes from the fact that one was defined using homology, the other using cohomology.)
Furthermore, the Hodge filtration of de Rham cohomology tells us which half of the extended period matrix to use.
```

### Differential forms of the second kind

The entries of the extended period matrix which do not come from ` are sometimes called `

*quasi-periods*.
We would like to be able to compute these as integrals too.
In order to do this, we introduce differential forms of the second kind
(the regular differential forms considered before are sometimes called *differential forms of the first kind*).

Differential forms of the second kind are a kind of meromorphic differential forms.
A *meromorphic differential form* is a differential forms which, locally at every point, looks like
```
where
```

` are local coordinates and `

` are meromorphic functions of `

```
.
(Meromorphic may be understood in terms of complex analysis or algebraic geometry as the context demands.)
In other words, meromorphic differential forms are differential forms which are allowed to have poles.
```

If is a meromorphic function, then is a meromorphic differential forms.
We call such forms *exact meromorphic forms*.

A *differential form of the second kind* is defined to be a meromorphic differential form which, locally at every point, is exact.
In other words, the poles of a differential form of the second kind must "look like" poles of exact meromorphic forms.

For a more concrete description of differential forms of the second kind, let's restrict attention to curves.
If is a differential form on a curve , we define the *residue* of at a point ```
by choosing a local coordinate at , writing out the Laurent expansion
```

```
and setting
```

Observe that an exact meromorphic form has residue zero at every point (because, in order to get to appear in ```
, you would have to differentiate but that is not meromorphic) and hence a differential form of the second kind has residue zero at every point.
It turns out that this is the only restriction: a meromorphic differential form on a curve is of the second kind if and only if it has residue zero at every point.
```

An example of a differential form of the second kind on an elliptic curve which is not regular or exact is .
This has a pole at infinity and nowhere else, and in terms of a suitable local coordinate at infinity it can be written as ` + something regular, so it has residue zero.`

### Computing the quasi-periods

The importance of differential forms of the second kind lies in the fact that ```
is isomorphic to the quotient
```

Hence we can compute quasi-periods by integrating closed differential forms of the second kind in the same way as we compute periods by integrating regular differential forms.

For example, if ` is the elliptic curve with equation `

`, then the forms `

` and `

` form a basis for `

```
and we can calculate the quasi-periods as
```

### The Legendre period relation for elliptic curves

I have already mentioned that an elliptic curve has periods ` and `

` obtained by integrating along generators for `

` and quasi-periods `

` and `

```
obtained by integrating along the same paths.
The Legendre period relation tells us that the extended period matrix has determinant , that is,
```

```
This post has got long enough without discussing that, so I will talk about it in the next post.
```