## 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.`