Martin's Blog

The Masser-Wüstholz isogeny theorem

Posted by martin 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 &endash; 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.

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The Masser-Wüstholz period theorem

Posted by martin 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 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 understand correctly 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 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.

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The Faltings height of an abelian variety over the rationals

Posted by martin 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 $\mathbb{Q}$, 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 $A$ over $\mathbb{Q}$, the Faltings height is the (logarithm of the) volume of $A$ as a complex manifold with respect to a particular volume form, chosen using the $\mathbb{Q}$-structure of $A$. 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 $\mathcal{A}_g$. Then he shows that the Faltings height is bounded within an isogeny class. Both of these parts are difficult.

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Shafarevich and Siegel's theorems

Posted by martin 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 $K$ be a number field and $S$ a finite set of places of $K$. Then there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$.

Siegel’s Theorem. Let $K$ be a number field and $S$ a finite set of places of $K$. An absolutely irreducible affine curve $C$ over $K$ of genus at least $1$ has only finitely many $S$-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.

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Finiteness theorems for abelian varieties

Posted by martin 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 $K$ and an abelian variety $A$ defined over $K$, there are only finitely many isomorphism classes of abelian varieties defined over $K$ and isogenous to $A$.

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Weil pairings: the skew-symmetric pairing

Posted by martin on Tuesday, 06 September 2011 at 13:52

Last time, we defined a pairing $ e_\ell : T_\ell A \times T_\ell (A^\vee) \to \lim_\leftarrow \mu_{\ell^n}. $ By composing this with a polarisation, we get a pairing of $T_\ell A$ with itself. This pairing is symplectic; the proof of this will occupy most of the post.

We will also see that the action of the Galois group on this pairing is given by the cyclotomic character, as I promised a long time ago. This tells us that the image of the $\ell$-adic Galois representation of $A$ is contained in $\operatorname{GSp}_{2g}(\mathbb{Q}_\ell)$. This is the end of my series on Mumford-Tate groups and $\ell$-adic representations attached to abelian varieties.

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Weil pairings: definition

Posted by martin on Monday, 29 August 2011 at 17:27

Recall that for an abelian variety $A$ over the complex numbers, $H_1(A^\vee, \mathbb{Z})$ is dual to $H_1(A, \mathbb{Z})$ (this is built in to the analytic definition of $A^\vee$). Since $T_\ell A \cong H_1(A, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{Z}_\ell$, this tells us that $T_\ell(A^\vee)$ is dual to $T_\ell A$ (as $\mathbb{Z}_\ell$-modules). We would like to show that this is true over other fields as well, which we will do by constructing the Weil pairings.

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Dual varieties over general fields

Posted by martin on Friday, 24 June 2011 at 17:26

Today we will construct dual abelian varieties over number fields. We use the universal property from two posts ago to define dual abelian varieties, then we give a simple construction inspired by the complex case. Proving that this construction satisfies the universal property is harder; in the case of number fields, we will use Galois descent to deduce it from the complex case which we already know analytically.

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Line bundles and morphisms to the dual variety

Posted by martin on Saturday, 28 May 2011 at 15:25

Over the complex numbers, the dual of an abelian variety $A$ is defined to have a Hodge structure dual to that of $A$. Hence morphisms $A \to A^\vee$ can be interpreted as bilinear forms on the Hodge structure of $A$. Of particular importance are the morphisms corresponding to Hodge symplectic forms.

Last time we saw that $A^\vee$ can also be interpreted as a group of line bundles on $A$. Today we will use this interpretation to define morphisms $\phi_\mathcal{L} : A \to A^\vee$ which turn out to be the same as those corresponding to Hodge symplectic forms. Then we generalise the definition of $\phi_\mathcal{L}$ to base fields other than $\mathbb{C}$, which we will use next time in constructing dual abelian varieties over number fields.

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