We start with an overview of heights on projective spaces and varieties to give a hint about their role in attacking finiteness problems of abelian varieties. We then try to explain the motivation for introducing the Faltings height on abelian varieties and do explicit computation in the case of elliptic curves. Finally we include a direct proof of the finiteness theorems of elliptic curves. Our main sources are ,  and . See also  and . This is a note prepared for the Faltings' Theorem seminar at Harvard.
Let us start by reviewing several basic properties of heights on projective spaces. The general scheme of height functions is a measurement of "arithmetic complexity". For a rational number , we can write it as for two integers without common divisors, and define the height of as It will be troublesome to draw a graph of this "function" defined on . Nevertheless, the definition matches our intuition: the larger is, the more complicated is.
More intrinsically, let be a number field. Set the absolute values:
Note that by the product formula, for , thus we know that does not depend on the choice of homogeneous coordinates . Because for , if and only if , it is easy to check that
This definition of heights extends to projective spaces of dimension in an obvious way.
The following is a prototype of a finiteness result under the bounded height condition.
Here comes a neat corollary of the Northcott's property.
Slightly more generally, given a projective variety , the embedding associates with every point a height using the height on the projective space . The height thus obtained does depend on the choice of the projective embedding, nevertheless, it turns out to be uniquely determined up to a bounded function. From the previous theorem, we know that over a number field , there are only finitely many points in with bounded heights.
One of the key steps in proving Faltings' theorem is to prove the finiteness theorems of abelian varieties.
Carl has showed the implication So it remains to show Finiteness I. Faltings' argument involves the usage of "heights":
Height I There are only finitely many isomorphism classes of polarized abelian varieties over of dimension , with semistable reduction everywhere and bounded "heights".
Height II The "height" is bounded in every isogeny class of abelian varieties over .
Assuming these two parts, then together with the semistable reduction theorem (every abelian variety has semistable reduction after a finite extension), we can easily deduce Finiteness I.
To possibly show the finiteness statement like Height I, we would like to associate a height to each abelian variety using Northcott's property. A natural option is to view an abelian variety as a point in the Siegel moduli variety and attach the height of that point to the corresponding abelian varieties. This motivates the notion of modular heights.
To clarify the proof, we recall the following lemma without proof.
It remains to construct such an . Because , have semistable reduction everywhere and they are isomorphic over a finite extension of , we know that , have the same set of places of bad reduction. Fix a prime , then is an extension of degree and is unramified outside by Neron-Ogg-Shafarevich's criterion. Therefore by Hermite's theorem, the compositum field of all 's must be a finite extension of . We claim that all 's are isomorphic to over . Let be an isomorphism over . Then for any , is an automorphism of which leaves fixed, therefore is the identity by the previous lemma. So the isomorphism is actually defined over . ¡õ
It would be wonderful to show Height II for the modular height, unfortunately, it is not clear how changes under isogeny. Faltings introduced what is now known as the Faltings height to attack Finiteness II. It turns out miraculously that the Faltings height can be proved to change only slightly under isogeny, and thus Height II is true for . More precisely,
Finally, to combine the Height I result for and Height II result for , one needs a comparison theorem between and : the boundedness of one of them implies the boundedness of the other.
Now the road-map for proving Finiteness I is
Height II (using the results of Raynaud and Tate on -divisible groups) and the comparison theorem (using the compactified Siegel modular variety over ) are the hardest parts of the whole proof and will occupy most of the remaining semester. In the rest of this talk, I will prove the comparison theorem for the case of elliptic curves, to somehow convince you that it is a reasonable thing to expect. If time permits, I will show Finiteness I for elliptic curves using a different argument, taking advantage of Siegel's theorem on integral points on elliptic curves.
We shall now motivate the definition of the Faltings height, which already showed up in Dick's introduction and also in Carl's talk. Suppose we have a complex elliptic curve for some period lattice . Intuitively, is more complicated if is more complicated, so we may attempt to define the height of as where is the fundamental domain for . However, this quantity is not well defined for a given isomorphism class: for example, scaling gives isomorphic elliptic curves, but is different. Notice that fixing the period lattice is equivalent to fixing a canonical choice for a differential : If is defined over , this canonical choice can be made using the minimal Weierstrass equation The differential is then well defined up to multiplication by and the period lattice is uniquely determined by the isomorphism class of . In this case,
We are using the crucial fact that is a PID to define the minimal Weierstrass equation. In general, for defined over a number field , its ring of integers is not necessarily a PID and a minimal Weierstrass equation does not exist. To obtain a canonical choice of the differential, we need the Neron models George talked about last time. Let be the Neron model of . Then the sheaf of Neron differentials is locally free of rank 1. So the pull back by the zero section gives us a projective - module of rank 1. Because is a PID, this module is actually a free module of rank 1. Therefore it has a canonical generator up to sign, which is exactly the above differential . For a number field , the same construction gives a projective -module of rank 1, where is the ring of integers of . When is not a PID, is not necessarily free and we cannot take a global generator, but only a bunch of local generators for each finite place. This motivates us to define the notion of metrized line bundles, introduced by Arakelov.
Now let us come back to the case of abelian varieties. Let be an abelian variety of dimension over . Let be the Neron model of . Then the sheaf of Neron differentials is locally free of rank on . So the top wedge power is a locally free sheaf of rank 1 on . Pulling back by the zero section , we obtain a line bundle . We specify the norm for every by
We first give an explicit formula of the Faltings height .
Using the previous explicit expression, now we can prove the comparison theorem of the Faltings height and the modular height for elliptic curves.
Since has semistable reduction everywhere, we know that if and only if dividing and in this case . Thus So it remains to show that which I shall leave it as an exercise using the arithmetic-geometric mean inequality. ¡õ
Finally, we shall utilize Siegel's theorem on the integral points of elliptic curves to give a completely different direct proof of Finiteness I for elliptic curves.
Siegel's proof uses techniques from Diophantine approximations, which we do not get into here. We will deduce Finiteness I from the even stronger Shafarevich's theorem for elliptic curves. The following cute proof is due to Shafarevich.
Heights and elliptic curves, Arithmetic geometry (Storrs, Conn., 1984), Springer, 1986, 253--265.
Abelian Varieties (v2.00), Available at www.jmilne.org/math/.
The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics), Springer, 2010.
Heights in Diophantine Geometry (New Mathematical Monographs), Cambridge University Press, 2007.
Diophantine Geometry: An Introduction (Graduate Texts in Mathematics), Springer, 2000.