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

and an abelian variety

defined over

,
there are only finitely many isomorphism classes of abelian varieties defined over

and isogenous to

.

Tags
abelian-varieties, alg-geom, faltings, maths, number-theory
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Posted by Martin Orr on
Tuesday, 06 September 2011 at 13:52

Last time, we defined a pairing

By composing this with a polarisation, we get a pairing of

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

-adic Galois representation of

is contained in

.
This is the end of my series on Mumford-Tate groups and

-adic representations attached to abelian varieties.

Tags
abelian-varieties, alg-geom, hodge, maths, number-theory
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Posted by Martin Orr on
Sunday, 21 November 2010 at 17:32

I said after my last post that I would write something about

-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

.
Next time I will relate this to Mumford-Tate groups.

Tags
abelian-varieties, alg-geom, maths, number-theory
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