Martin Orr's Blog

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 A whose tangent space contains a given period \omega. 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 T_0 A. In particular \omega 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 \omega. Several applications of the Strong Proposition with different hyperplanes in T_0 A prove the Period Theorem.

no comments Tags abelian-varieties, alg-geom, maths, number-theory Read more...

The matrix lemma for elliptic curves

Posted by Martin Orr on Friday, 25 May 2012 at 14:00

Let A be a principally polarised abelian variety of dimension g over \mathbb{C}. We can associate with A a g \times g complex matrix called the period matrix which roughly speaking describes a basis for the image of H_1(A, \mathbb{Z}) in T_0 A(\mathbb{C}) (actually it is not really the period matrix as it is only defined up to the action of \mathop{\mathrm{Sp}}_{2g}(\mathbb{Z}) on the Siegel upper half space; we can make it nearly unique by forcing it to be in a particular fundamental domain).

The matrix lemma says that, if A is defined over a number field, then the entries of the imaginary part of the period matrix cannot be too large with respect to the height of A (Faltings height or modular height).

Matrix lemma. (Masser 1987) Let A be a principally polarised abelian variety of dimension g over a number field K. Let \tau be the period matrix for A in the standard fundamental domain of the Siegel upper half space. There is a constant c depending only on g such that all the entries of \tau satisfy \lvert \mathop{\mathrm{Im}} \tau_{ij} \rvert \leq c [K:\mathbb{Q}] \max(1, h(A)).

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.

1 comment Tags abelian-varieties, alg-geom, maths, number-theory Read more...

The Masser-Wüstholz isogeny theorem

Posted by Martin Orr on Wednesday, 25 April 2012 at 14:09

Let A and B be two isogenous abelian varieties over a number field K. Can we be sure that there is an isogeny between them of small degree, where "small" is an explicit function of A and K? In particular, our bound should not depend on B; 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 A and B be principally polarised abelian varieties over a number field K. Suppose that there exists some isogeny A \to B. Then there is an isogeny A \to B of degree at most c \max([K:\mathbb{Q}], h(A))^\kappa where c and \kappa are constants depending only on the dimension of A.

We will prove this using the Masser-Wüstholz period theorem which I discussed last time.

2 comments Tags abelian-varieties, alg-geom, maths, number-theory Read more...

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 A having a given period of A 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 A be an abelian variety defined over a number field k with a principal polarisation \lambda. For any non-zero period \omega of A, the smallest abelian subvariety A_\omega of A whose tangent space contains \omega satisfies \deg_\lambda A_\omega \leq C \max([k:\mathbb{Q}], h_F(A), H_\lambda(\omega, \omega))^\kappa where C and \kappa are constants depending only on \dim A.

Masser and Wüstholz gave a value for \kappa of (g-1) 4^g g! where g = \dim A. 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 \kappa = 3g + \epsilon suffices.

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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 \mathbb{Q} only, and we used two properties of \mathbb{Q}: 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.

2 comments Tags abelian-varieties, alg-geom, faltings, maths, number-theory Read more...

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