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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...

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