## 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.