Martin Orr's Blog

The Faltings height of an abelian variety over the rationals

Posted by Martin Orr 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.

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

Siegel's theorem for curves of genus 0

Posted by Martin Orr on Friday, 28 October 2011 at 12:35

Last time we proved Siegel's theorem on the finiteness of integer points on affine curves of genus at least 1. The theorem applies also to curves of genus 0 with at least 3 points at infinity. I shall give a simple proof that deduces this from the higher genus case, then another proof using Baker's theorem from transcendental number theory which gives an effective bound on the heights of the points.

Theorem. Let K be a number field and S a finite set of places of K. Let X be an affine K-curve of genus 0 such that there are at least 3 \bar{K}-points in the projective closure of X which are not in X. Then X has finitely many S-integer points.

The condition that there should be at least 3 points at infinity is necessary: the affine line is a genus 0 curve with 1 point at infinity and infinitely many integer points, and the curve x^2 - Dy^2 = 1 for D a non-square positive integer has 2 points at infinity and infinitely many integer points.

2 comments Tags maths, number-theory Read more...

Shafarevich and Siegel's theorems

Posted by Martin Orr 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.

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

Finiteness theorems for abelian varieties

Posted by Martin Orr 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.

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

Weil pairings: the skew-symmetric pairing

Posted by Martin Orr 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 (inverse of the) cyclotomic character, as I promised a long time ago (in the comments). 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.

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

1 2 3 4 5 6 7 8 ... 12 13

Archives