## The Masser-Wüstholz isogeny theorem

Posted by Martin Orr on Wednesday, 25 April 2012 at 14:09

Let ` and `

` be two isogenous abelian varieties over a number field `

```
.
Can we be sure that there is an isogeny between them of small degree, where "small" is an explicit function of
```

` and `

```
?
In particular, our bound should not depend on
```

`; this means that the bound will imply Finiteness Theorem I, and hence the Shafarevich, Tate and Mordell conjectures.`

The Masser-Wüstholz isogeny theorem answers this question, at least subject to a minor condition on polarisations (I think that this was removed in a later paper of Masser and Wüstholz but it is not too important anyway -- when deducing Finiteness Theorem I you can remove the polarisation issue with Zarhin's Trick).

Theorem.(Masser, Wüstholz 1993) Let`and`

`be principally polarised abelian varieties over a number field`

`. Suppose that there exists some isogeny`

`. Then there is an isogeny`

`of degree at most`

`where`

`and`

`are constants depending only on the dimension of`

`.`

We will prove this using the Masser-Wüstholz period theorem which I discussed last time.

### Proof of the isogeny theorem

Recall from last time that in order to obtain an isogeny of bounded degree between a subvariety of ` and a subvariety of `

`, it suffices to have a non-split subvariety of bounded degree in `

```
.
The central idea of the proof is that we will use the period theorem, along with the assumption that
```

` and `

` are isogenous, to construct such a non-split subvariety.`

Actually we will construct a non-split subvariety ` of `

`, of degree `

```
say.
Once we have done this, it is easy to finish the proof:
by the lemma from last time, there are non-zero subvarieties
```

` and `

` and an isogeny `

` of degree at most `

`.`

If we suppose that ` is simple, then we must have `

` and at least one of the projections `

```
is an isogeny.
Because the degree of
```

` is bounded, so is the degree of `

` and hence `

` is the desired isogeny `

` of bounded degree.`

If ` is not simple, then we can do something similar to get an isogeny `

` where `

` and `

` of bounded degree; then we quotient out by `

` and `

` and induct.`

### Finding a non-split subvariety

It remains to show that for any isogenous abelian varieties ` and `

`, there is a non-split subvariety `

` with degree bounded polynomially by `

`, `

` and `

`.`

Choose a non-zero period ` and a basis `

` for `

```
.
Let
```

` be the period `

` of `

`.`

Let ` be the smallest abelian subvariety of `

` whose tangent space contains `

```
.
Now the period theorem gives a bound for the degree of
```

`, and the assumption that `

` is isogenous to `

` implies that `

` is non-split.`

To prove the latter, let ` be any isogeny `

```
.
There are integers
```

```
such that
```

```
Then
```

```
must be contained in
```

```
Any subvariety of
```

` with a non-trivial projection to `

` is non-split.`

(Observe that the degree of ` may be really big, and the degree of `

` is related to the degree of `

`, so the degree of `

```
may be big.
But this is OK because we are not using
```

` to say anything about the degree of `

` - that comes from the period theorem.)`

### Bounding the degree of the subvariety

The period theorem tells us that
```
where
```

` and `

` are principal polarisations of `

` and `

`, and `

` is the Hermitian form associated with the polarisation `

` of `

`.`

We want to remove ```
from this bound, by bounding it in terms of the other quantities.
Let us write
```

```
This is a norm on the real vector space
```

`.`

We have
```
so it will suffice to show that we can choose
```

` and `

` with bounded lengths.`

The only condition that we have put on ` is that it is a period i.e. in the lattice `

```
.
A fundamental domain for this lattice has volume 1 with respect to our chosen metric (this is equivalent to the polarisation
```

` being principal) and so Minkowski's theorem gives an upper bound for the length of the smallest period, depending only on the dimension `

```
:
```

With regard to `, they must be a basis for `

```
.
Again this lattice has covolume 1 and so by Minkowski's theorem, there is a basis such that
```

```
for a constant
```

` depending only on `

`.`

An upper bound for the product ` does not imply an upper bound for the individual lengths (or for `

`) but we can deduce such a bound if we also have a `

*lower* bound for the ```
.
The following such bound can be deduced from a refinement of Masser's Matrix Lemma.
```

Lemma.Let`be a principally polarised abelian variety defined over a number field`

`. There is a constant`

`depending only on`

`such that every non-zero period`

`of`

`satisfies`

### Finishing the proof

Combining the above, we get that the isogeny ```
of least degree satisfies
```

```
(The constants
```

` and `

` are different in different inequalities in this post.)`

However this bound depends on ```
, which we wanted to avoid.
We use a basic fact about the Faltings height: if there is an isogeny
```

```
, then
```

```
So we get that
```

```
It might seem silly that we want to bound
```

` and we have just introduced it on the right hand side, but it is only a polynomial in `

` so with a bit of rearrangement we can absorb it into the constant.`

Barinder Banwaitsaid on Friday, 27 April 2012 at 18:55 :"Any subvariety of

`with a non-trivial projection to`

`is non-split". Why is this? Are you using that`

`is simple?`

Right at the end, "So we get that...": Are you claiming that

`?`

Martin Orrsaid on Wednesday, 02 May 2012 at 08:53 :No. I use that

`so any split subvariety contained in`

`is in fact contained in`

`. Since`

`is finite, the subvariety is contained in`

`i.e. has trivial projection to`

`.`

Again no. I confusingly left out some steps. We use the fact that

`so`

`then we can hide the constants in the big constant`

`.`