# Martin's Blog

## 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.