# Martin's Blog

## The Masser-Wüstholz period theorem

Posted by Martin Orr on Friday, 30 March 2012 at 12:20

I wanted to write a post about the Masser-Wüstholz isogeny theorem, which gives a quantitative version of Finiteness Theorem I. But it turned out to be too long so for today I will focus on the main ingredient in the proof of the isogeny theorem: the Masser-Wüstholz period theorem.

The period theorem gives a bound for the degree of the smallest abelian subvariety of a fixed abelian variety having a given period of in its tangent space. In this post I will explain the statement of the period theorem, in particular defining the degree of a (polarised) abelian variety, and give some properties of the degree which will be used in the proof of the isogeny theorem.

Period Theorem. (Masser, Wüstholz 1993) Let be an abelian variety defined over a number field with a principal polarisation . For any non-zero period of , the smallest abelian subvariety of whose tangent space contains satisfies where and are constants depending only on .

Masser and Wüstholz gave a value for of where . For myself, I am only interested in the existence of such a bound, but work has been done on improving it. If I correctly understand a recent preprint of Gaudron and Rémond, they show that suffices.

### Periods

Let be an abelian variety of dimension .

A period of is an element of the kernel of the exponential map . Let denote this kernel; recall that is a lattice in the complex vector space .

One of the ways of interpreting a polarisation is as a positive definite Hermitian form on . This gives us the quantity on the right hand side of the period theorem.

### The degree of a polarised abelian variety

Let be an abelian variety of dimension and a polarisation of . There are two notions of degree of , which differ by a factor of .

First we define what Masser and Wüstholz call the normalised degree . This is the volume of a fundamental domain for in , measured using the norm .

An equivalent way of defining the normalised degree is as the square root of the degree of , viewed as an isogeny .

The unnormalised degree comes from interpreting the polarisation as an ample line bundle on . Intersection theory gives a general notion of the degree of any line bundle on a projective variety.

The two degrees are related by the Riemann-Roch theorem for abelian varieties:

For our purposes it will be convenient to work always with the normalised degree. I stated the period theorem with the unnormalised degree because that is the standard statement, but of course we can replace it by the normalised degree if we want.

### The degree of the graph of an isogeny

Let and be two polarised abelian varieties. For the rest of this post we will consider degrees of subvarieties of . I shall write with no subscript to mean the normalised degree of polarised by the restriction of .

Note that , and (the last relation would not be true with unnormalised degrees - there is a factor depending on the dimensions).

Suppose that there is an isogeny of degree . Then the graph of is an abelian subvariety of . We can bound its degree from below by the degree of the isogeny .

Lemma.

Proof. Let , be the projections and . Note that is an isomorphism and .

We get polarisations and on . Let , be the real parts of the associated Hermitian forms, which we may interpret as positive-definite symmetric matrices.

We have so it will suffice to prove that

Recall that means the degree of with respect to the restriction of , which is given by . So we have

Thus the lemma follows from the fact that for any positive-definite symmetric real matrices and . This inequality is left as an exercise for the reader.

### From subvarieties to isogenies

When can we reverse the process of going from an isogeny to its graph? That is, when does an abelian subvariety of tell us something about an isogeny? Some choices of subvariety such as are clearly of no use, so we introduce the following definition.

Definition. We say that a subvariety of is split if it has the form for some and .

Suppose that there is a non-split abelian subvariety . First if and are simple, then the projections and must be isogenies so and are isogenous.

We would like to show that there is an isogeny whose degree is bounded by a function of . There is an isogeny such that . Then , where .

By the above lemma, and are bounded above by so there is an isogeny of degree at most

Dropping the assumption that and are simple we can prove the following:

Lemma. If has a non-split abelian subvariety then there are non-zero abelian subvarieties and and an isogeny such that where .

1. The Masser-Wüstholz isogeny theorem From Martin's Blog

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 ...

2. Main steps of the proof of the period theorem From Martin's Blog

Today I will explain how to prove the Masser-Wüstholz Period Theorem starting from the Key Proposition, an weaker existence result for abelian subvarieties of bounded degrees. The Key Proposition and the Tangent Space Lemma, which I mention brief...