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.
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.
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 
Posted by Martin Orr on Friday, 25 May 2012 at 14:00
Let 
The matrix lemma says that, if
Matrix lemma. (Masser 1987) Let
be a principally polarised abelian variety of dimensionover a number field. Letbe the period matrix forin the standard fundamental domain of the Siegel upper half space. There is a constantdepending only onsuch that all the entries ofsatisfy
Last time I used a lower bound for the lengths of non-zero periods in the proof of the isogeny theorem. This follows from the matrix lemma as we can easily relate lengths of periods and the period matrix.
Today I will prove the matrix lemma for elliptic curves. The general proof requires various facts about Siegel modular forms and also uses a funny choice of level structure due to Igusa (I do not understand why). But the basic structure of the proof is already visible in the elliptic curves case and we can be concrete about the modular forms involved, using only facts I learned in Part III.
Posted by Martin Orr on Wednesday, 25 April 2012 at 14:09
Let 
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
andbe principally polarised abelian varieties over a number field. Suppose that there exists some isogeny. Then there is an isogenyof degree at mostwhereandare constants depending only on the dimension of.
We will prove this using the Masser-Wüstholz period theorem which I discussed last time.
![\lvert \mathop{\mathrm{Im}} \tau_{ij} \rvert \leq c [K:\mathbb{Q}] \max(1, h(A)).](http://www.martinorr.name/blog/images/mathtex/1168.png)
![c \max([K:\mathbb{Q}], h(A))^\kappa](http://www.martinorr.name/blog/images/mathtex/1196.png)
