Martin's Blog

The Masser-Wüstholz isogeny theorem

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

Let A and B be two isogenous abelian varieties over a number field K. Can we be sure that there is an isogeny between them of small degree, where "small" is an explicit function of A and K? In particular, our bound should not depend on B; 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 A and B be principally polarised abelian varieties over a number field K. Suppose that there exists some isogeny A \to B. Then there is an isogeny A \to B of degree at most  c \max([K:\mathbb{Q}], h(A))^\kappa where c and \kappa are constants depending only on the dimension of A.

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

2 comments Tags abelian-varieties, alg-geom, maths, number-theory