# 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.

## The Faltings height and normed modules

Posted by Martin Orr on Saturday, 31 December 2011 at 15:31

In this post I shall give the definition of the Faltings height of an abelian variety over any number field. Last time we did this over only, and we used two properties of : the integers are a PID and there is only one archimedean place. To do things more generally, we will introduce the technology of normed modules and their degrees.

## The Faltings height of an abelian variety over the rationals

Posted by Martin Orr on Thursday, 17 November 2011 at 15:58

The Faltings height is a real number attached to an abelian variety (defined over a number field), which is at the centre of Faltings' proof of Finiteness Theorem I. In this post all I will do is define the Faltings height of an abelian variety over , as already this requires a lot of preliminaries on cotangent and canonical sheaves of schemes. Further complications arise over other base fields, which I will discuss next time.

For an abelian variety over , the Faltings height is the (logarithm of the) volume of as a complex manifold with respect to a particular volume form, chosen using the -structure of . The preliminaries are needed in order to choose the volume form.

Faltings' proof of Finiteness I proceeds by showing that for any fixed number field, there are finitely many abelian varieties of bounded Faltings height. This is done by showing that the Faltings height is not far away from the classical height of a point representing the abelian variety in the moduli space . Then he shows that the Faltings height is bounded within an isogeny class. Both of these parts are difficult.

## Shafarevich and Siegel's theorems

Posted by Martin Orr on Friday, 07 October 2011 at 09:00

In this post I will prove the Shafarevich conjecture for elliptic curves (also called Shafarevich's theorem). The proof is by reducing it to the finiteness of the number of solutions of a certain Diophantine equation, and then applying Siegel's theorem on integral points on curves.

Shafarevich's Theorem. Let be a number field and a finite set of places of . Then there are only finitely many isomorphism classes of elliptic curves over with good reduction outside .

Siegel's Theorem. Let be a number field and a finite set of places of . An absolutely irreducible affine curve over of genus at least has only finitely many -integral points.

Since the reduction of Shafarevich's theorem to Siegel's theorem is short, and Siegel's theorem is of independent interest, most of the post will be about Siegel's theorem.

## Finiteness theorems for abelian varieties

Posted by Martin Orr on Monday, 19 September 2011 at 16:34

Faltings famously proved the Mordell, Shafarevich and Tate conjectures in 1983. In this post I will discuss the relationships between the Tate and Shafarevich conjectures and some other finiteness theorems for abelian varieties.

Everything which I call a conjecture in this post is known to be true: they all follow from Finiteness Theorem I. Proving Finiteness Theorem I was the bulk of Faltings' work, but I am not going to talk about that today.

Finiteness Theorem I. Given a number field and an abelian variety defined over , there are only finitely many isomorphism classes of abelian varieties defined over and isogenous to .