Posted by Martin Orr on
Wednesday, 05 December 2012 at 16:04

I intend to return to the basic theory of abelian varieties and write write a few posts on their endomorphism algebras and associated moduli spaces.
To begin with, I will discuss the Rosati involution which is an involution of the endomorphism algebra coming from a polarisation.
The existence of such an involution is crucial for the classification of endomorphism algebras which I will discuss next.

Tags
abelian-varieties, alg-geom, maths
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Posted by Martin Orr on
Thursday, 20 September 2012 at 17:39

I recently had to write an account of my research for a non-specialist. I think I took it more seriously than necessary -- I am not sure that anyone will read it. But it might interest some readers of this blog, so I shall put it here. I have just put the paper containing this work on Arxiv.

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alg-geom, maths, shimura-varieties
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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.

Tags
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.

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