# 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.

### 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.

1. Matrix lemma for elliptic curves From Martin's Blog

Let be a principally polarised abelian variety of dimension over . We can associate with a complex matrix called the period matrix which roughly speaking describes a basis for the image of in (actually it is not really the period matrix as it is o...

1. Barinder Banwait said on Friday, 27 April 2012 at 18:55 :
1. "Any subvariety of with a non-trivial projection to is non-split". Why is this? Are you using that is simple?

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

2. Martin Orr said on Wednesday, 02 May 2012 at 08:53 :
1. 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 .

2. Again no. I confusingly left out some steps. We use the fact that so then we can hide the constants in the big constant .