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.

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abelian-varieties, alg-geom, maths, number-theory
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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.

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abelian-varieties, alg-geom, maths, number-theory
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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.

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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.

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abelian-varieties, alg-geom, faltings, maths, number-theory
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