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Maths > Abelian varieties > The Masser-Wüstholz isogeny theorem

Main steps of the proof of the period theorem

Posted by Martin Orr on Friday, 03 August 2012 at 11:06

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 briefly, are the main results proved by Masser and Wüstholz by the techniques of transcendental number theory on their way to the Period Theorem.

Recall that the Period Theorem is a bound for the degree of the smallest abelian subvariety of whose tangent space contains a given period . In the Key Proposition we find a subvariety of bounded degree whose tangent space satisfies the very weak condition of being inside a chosen hyperplane in . In particular need not be in the tangent space of the subvariety. However we use the Key Proposition and induction to prove the Strong Proposition, which gives a subvariety whose tangent space does contain . Several applications of the Strong Proposition with different hyperplanes in prove the Period Theorem.

Heights of vector subspaces

Let be a vector space of dimension over a number field . We can identify the set of -dimensional subspaces of with the points of a projective variety, called the Grassmannian .

The simplest example of this is : by definition, points in correspond to lines in . Hence . For a recipe for how to do this in general, see Wikipedia.

We now define the height of a subspace to be the logarithmic Weil height of the corresponding point in the Grassmannian.

The Tangent Space Lemma

Let be an abelian variety of dimension defined over a number field . The tangent space to at the origin is then a -vector space of dimension . (This is the tangent space in the algebraic geometry sense. The complex analytic tangent space, which contains the period lattice, is .)

If is an abelian subvariety of also defined over , then is a -vector subspace of , and so has a height as defined above. Note that has nothing to do with , the Faltings height of . Indeed depends only on the isomorphism class of , while depends on how is embedded as a subvariety of .

One of the two main results proved by Masser and Wüstholz on the way to the Period Theorem is the Tangent Space Lemma, which bounds in terms of the degree of .

Tangent Space Lemma. Let be a principally polarised abelian variety defined over a number field and an abelian subvariety. There exists a constant depending only on such that

The Strong Proposition

The other main ingredient in the proof of the Period Theorem is the Key Proposition. This is an existence result for subvarieties of of bounded degree, although the subvariety satisfies only a very weak condition on its tangent space -- in particular might not contain the period .

Key Proposition. Let be a polarised abelian variety and an abelian subvariety, defined over a number field . Let be a hyperplane in defined over , and suppose that contains a non-zero period of .

Then there exists a non-zero abelian subvariety of such that and where , is a constant depending only on , and .

Note that this is essentially a proposition about , with playing little role in the conclusion. The only benefit of allowing is that we are going to apply the proposition to various different subvarieties of a fixed , and we want the bound to always depend on the height of rather than on the varying height of .

We will use the Key Proposition to prove the Strong Proposition. This is the same statement as the Key Proposition, with the extra condition that and with the exponent in the bound replaced by some function of . The implication Key Proposition => Strong Proposition is proved in the next section.

To prove the Period Theorem from the Strong Proposition: Let . We can choose hyperplanes in such that Furthermore we can choose the such that for all .

Applying the Strong Proposition to each , with , we get abelian subvarieties of bounded degree such that and . The first condition implies that and the second that . Hence is the identity connected component of , which implies that

The Strong Proposition gives us bounds for each . We just have to get rid of the term from these bounds. But by the Tangent Space Lemma, This gives and we can get rid of the by increasing the constant, completing the proof of the Period Theorem.

From Key Proposition to Strong Proposition

The Strong Proposition is deduced from the Key Proposition by induction on . We begin by applying the Key Proposition directly to the given and , getting such that .

If then we can stop. In particular this always happens if since then so that . This gives the base case of the induction.

Otherwise, let be the orthogonal complement of in . This is the abelian subvariety of such that and are orthogonal with respect to .

The tangent space of is the direct sum of the tangent spaces of and . This need not be true for the period lattices: the sum of the period lattices of and has finite index in the period lattice of . Let this index be ; we can prove that .

Now has codimension 1 in so we can apply the Strong Proposition inductively to and . To do this we need a period of contained in . We can write with and periods of and respectively, and use .

So applying the Strong Proposition inductively gives us an abelian subvariety such that where

By a Theorem of Schmidt, Struppeck and Vaaler, and by the Tangent Space Lemma, We can also show that so the above bound simplifies to

Finally the abelian subvariety satisfies the conditions of the Strong Proposition (its tangent space is contained in and contains ), and its degree satisfies Using the above bounds for and the bound for from the Key Proposition completes the proof of the Strong Proposition.