## The Masser-Wüstholz period theorem

Posted by Martin Orr on Friday, 30 March 2012 at 12:20

I wanted to write a post about the Masser-Wüstholz isogeny theorem, which gives a quantitative version of Finiteness Theorem I. But it turned out to be too long so for today I will focus on the main ingredient in the proof of the isogeny theorem: the Masser-Wüstholz period theorem.

The period theorem gives a bound for the degree of the smallest abelian subvariety of a fixed abelian variety ` having a given period of `

```
in its tangent space.
In this post I will explain the statement of the period theorem, in particular defining the degree of a (polarised) abelian variety, and give some properties of the degree which will be used in the proof of the isogeny theorem.
```

Period Theorem.(Masser, Wüstholz 1993) Let`be an abelian variety defined over a number field`

`with a principal polarisation`

`. For any non-zero period`

`of`

`, the smallest abelian subvariety`

`of`

`whose tangent space contains`

`satisfies`

`where`

`and`

`are constants depending only on`

`.`

Masser and Wüstholz gave a value for ` of `

` where `

```
.
For myself, I am only interested in the existence of such a bound, but work has been done on improving it.
If I correctly understand a recent preprint of Gaudron and Rémond, they show that
```

` suffices.`

### Periods

Let ` be an abelian variety of dimension `

`.`

A *period* of ` is an element of the kernel of the exponential map `

```
.
Let
```

` denote this kernel; recall that `

` is a lattice in the complex vector space `

`.`

One of the ways of interpreting a polarisation is as a positive definite Hermitian form ` on `

```
.
This gives us the quantity
```

` on the right hand side of the period theorem.`

### The degree of a polarised abelian variety

Let ` be an abelian variety of dimension `

` and `

` a polarisation of `

```
.
There are two notions of degree of
```

`, which differ by a factor of `

`.`

First we define what Masser and Wüstholz call the *normalised degree* ```
.
This is the volume of a fundamental domain for
```

` in `

```
,
measured using the norm
```

`.`

An equivalent way of defining the normalised degree is as the square root of the degree of `, viewed as an isogeny `

`.`

The *unnormalised degree* ` comes from interpreting the polarisation as an ample line bundle on `

```
.
Intersection theory gives a general notion of the degree of any line bundle on a projective variety.
```

The two degrees are related by the Riemann-Roch theorem for abelian varieties:

For our purposes it will be convenient to work always with the normalised degree. I stated the period theorem with the unnormalised degree because that is the standard statement, but of course we can replace it by the normalised degree if we want.

### The degree of the graph of an isogeny

Let ` and `

```
be two polarised abelian varieties.
For the rest of this post we will consider degrees of subvarieties
```

` of `

```
.
I shall write
```

` with no subscript to mean the normalised degree of `

` polarised by the restriction of `

`.`

Note that `, `

` and `

` (the last relation would not be true with unnormalised degrees - there is a factor depending on the dimensions).`

Suppose that there is an isogeny ` of degree `

```
.
Then the graph of
```

` is an abelian subvariety `

` of `

```
.
We can bound its degree from below by the degree of the isogeny
```

`.`

Lemma.

Proof.Let`,`

`be the projections`

`and`

`. Note that`

`is an isomorphism and`

`.`

We get polarisations

`and`

`on`

`. Let`

`,`

`be the real parts of the associated Hermitian forms, which we may interpret as positive-definite symmetric matrices.`

We have

`so it will suffice to prove that`

Recall that

`means the degree of`

`with respect to the restriction of`

`, which is given by`

`. So we have`

Thus the lemma follows from the fact that

`for any positive-definite symmetric real matrices`

`and`

`. This inequality is left as an exercise for the reader.`

### From subvarieties to isogenies

When can we reverse the process of going from an isogeny to its graph?
That is, when does an abelian subvariety of ```
tell us something about an isogeny?
Some choices of subvariety such as
```

` are clearly of no use, so we introduce the following definition.`

Definition.We say that a subvariety of`is`

splitif it has the form`for some`

`and`

`.`

Suppose that there is a non-split abelian subvariety ```
.
First if
```

` and `

` are simple, then the projections `

` and `

` must be isogenies so `

` and `

` are isogenous.`

We would like to show that there is an isogeny ` whose degree is bounded by a function of `

```
.
There is an isogeny
```

` such that `

```
.
Then
```

`, where `

`.`

By the above lemma, ` and `

` are bounded above by `

` so there is an isogeny `

```
of degree at most
```

Dropping the assumption that ` and `

` are simple we can prove the following:`

Lemma.If`has a non-split abelian subvariety`

`then there are non-zero abelian subvarieties`

`and`

`and an isogeny`

`such that`

`where`

`.`