Martin's Blog

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...

Weil pairings: definition

Posted by Martin Orr 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.

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

Tate modules

Posted by Martin Orr on Sunday, 21 November 2010 at 17:32

I said after my last post that I would write something about \ell-adic representations coming from abelian varieties. I have finally got around to doing so: here I will tell the story of how these representations are defined, and show that the Tate module is canonically isomorphic to H_1(A, \mathbb{Z}) \otimes \mathbb{Z}_\ell. Next time I will relate this to Mumford-Tate groups.

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

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