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 
only, and we used two properties of : 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.
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 , 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 
over , the Faltings height is the (logarithm of the) volume of as a complex manifold with respect to a particular volume form, chosen using the -structure of .
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 
.
Then he shows that the Faltings height is bounded within an isogeny class.
Both of these parts are difficult.
Posted by martin 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
be a number field and
a finite set of places of
be an affine
-points in the projective closure of
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 
for 
a non-square positive integer has 2 points at infinity and infinitely many integer points.
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
Siegel’s Theorem. Let
over
has only finitely many
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.
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
Posted by martin 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 cyclotomic character,
as I promised a long time ago.
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.
Posted by martin on Monday, 29 August 2011 at 17:27
Recall that for an abelian variety over the complex numbers,
is dual to 
(this is built in to the analytic definition of 
).
Since 
,
this tells us that 
is dual to (as 
-modules).
We would like to show that this is true over other fields as well,
which we will do by constructing the Weil pairings.
Posted by martin 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.
Posted by martin on Monday, 05 April 2010 at 22:34
An algebraic function is a function which we obtain by solving a polynomial in two variables 
and 
to write as a function of . In general polynomials have more than one root, so (informally) we get a multi-valued function. In this post I will restrict attention to regions of the plane in which we can unambiguously pick a single “branch” of the function. What happens where branches meet will be the subject of a later post.
I shall give an algorithm for expressing an algebraic function (in such a nicely behaved region) as a power series, thereby proving that a power series solution to the original polynomial exists. The generalisation of this result to a complete discrete valuation ring is Hensel’s lemma, and is particularly important to number theorists in the case of p-adic integers (which were invented by Hensel). In this post I will focus on the case of algebraic functions, as it is easier to apply geometric intuition.