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 [1], [2] and [3]. See also [4] and [5]. This is a note prepared for the Faltings' Theorem seminar at Harvard.

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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:

- satisfies for and is the real or complex absolute value for real or complex.
- , where . It takes the value on the uniformizer .

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

Proposition 1
If an element generates the ideal , where are relative prime integral ideals of , then

This definition of heights extends to projective spaces of dimension in an obvious way.

Definition 2
Let . We define the *relative height* the *absolute height* and the *logarithmic height* Note that for a finite extension , , therefore the absolute height and the logarithmic height do not depend on the choice of .

The following is a prototype of a finiteness result under the bounded height condition.

Theorem 1 (Northcott)
For any constants and , there are only finitely many points satisfying and . In particular, for any number field , there are only finitely many points with bounded height.

Proof
Suppose and let be the Galois conjugates of . Then the minimal polynomial of is of the form Note that the heights of the 's are all the same, therefore the coefficient of is bounded by for some constant not depending on . So these coefficients are rational numbers with bounded heights, therefore there are only finitely many choices of the coefficients, hence finitely many choices of .
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Here comes a neat corollary of the Northcott's property.

Proof
Suppose for some . Then , hence . Conversely, suppose , then for every . By the Northcott's, some and have to be same, which implies is a root of unity.
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Remark 2
Though we do not need it in this talk, it does not harm to mention that there is a similar notion , called the *canonical height* (or *Neron-Tate height*) attached to every rational point on an abelian variety over . It measures the arithmetic complexity of the point: for example, if and only if is a torsion point analogously.

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.

Theorem 2 (Finiteness I, or Conjecture T)
Let be an abelian variety over a number field . Then there are only finitely many isomorphism classes of abelian varieties over isogenous to .

Theorem 3 (Finiteness II, or Shafarevich's conjecture for abelian varieties)
There are only finitely many isomorphism classes of abelian varieties over of dimension having good reduction outside a finite set of places .

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.

Definition 3
Let be a polarized abelian variety over of dimension and degree . Let be the Siegel modular variety with its canonical projective embedding. Then associated with we have a point .We define the *modular height* of to be .

Theorem 4 (Height I)
Let be a constant. Then there are only finitely many isomorphism classes of polarized abelian varieties over of dimension , degree having semistable reduction everywhere and .

Remark 4
This is not true without the semistable reduction condition: for any , are all isomorphic over but not isomorphic over . These 's have the same -invariant, hence are the same.

To clarify the proof, we recall the following lemma without proof.

Lemma 1
Let be a polarized abelian variety. Then

- The group of automorphisms is finite.
- Suppose , an automorphism of acting on trivially must be the identity.

Proof (Height I)
By Northcott's property, we know that the set of -isomorphism classes of such is finite. So we need to show that given a polarized abelian variety with semistable reduction everywhere, there are only finitely many -isomorphism classes with semistable reduction everywhere which are isomorphic to over . We shall show that there exists a finite extension such that all these are actually isomorphic to over . Then 's are parametrized by , which is finite, since and (by the previous lemma) are finite. This completes the 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,

Theorem 5 (Height II)
Let be an abelian variety over having semistable reduction everywhere. Then is bounded in the isogeny class of .

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.

Theorem 6 (Comparison of heights)
There exists constants , , such that for abelian varieties over with semistable reduction everywhere,

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.

Definition 4
Let be the ring of integers of . A *metrized line bundle* on is a pair , where is a line bundle on (i.e. a projective -module of rank 1) and for each , is a norm on the real or complex vector space .

Definition 5
For , suppose , we set if for . For , we denote . The *height* of a metrized line bundle is defined to be for any . It is well-defined by the product formula. The *degree* of is defined to be

Remark 5
The nonarchimedean part is equal to . When and is the generator of , one then recovers that (one thinks of as the area ).

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

Definition 6
Let be the metrized line bundle on described above. The *Faltings height* of is defined to be

Remark 6
Since the Neron model may change after field extension, the Faltings height *depends* on the base field . However, by the semistable stable reduction theorem, the Neron model does not change after base change to some finite field extension. We thus define the *stable Faltings height* to be the Faltings height , for any such that has semistable stable reduction everywhere.

We first give an explicit formula of the Faltings height .

Theorem 7
Let be an elliptic curve. Suppose for . Then where is the minimal discriminant and is the modular discriminant function.

Proof
Let be any Weierstrass equation of . We shall utilize the invariance of the section of under the change of coordinates to calculate the Faltings height locally. For , let be the Neron differential at , then by the invariance of , we know that where is the minimal discriminant of at . So for , Hence the local contribution at is and the total nonarchimedean contribution is . For , let be the Weierstrass equation given by and then by the invariance of , we know that We compute therefore the local contribution at is and the total archimedean contribution is This completes the proof.
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Using the previous explicit expression, now we can prove the comparison theorem of the Faltings height and the modular height for elliptic curves.

Theorem 8
There exists some constant such that for elliptic curves with semistable reduction everywhere,

Proof
For any , we can suitably choose such that , so that . Hence implies that Using the -expansion of , one also knows that Therefore Also from we obtain Plugging into the explicit formula of the Faltings height gives

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.

Theorem 9 (Siegel)
Let be an (affine) elliptic curve, be a finite set containing and be the ring of -integers. Then the set of integral points is finite.

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.

Theorem 10 (Shafarevich, Finiteness II)
Let be a finite set containing . Then there are only finitely many isomorphism classes having good reduction outside .

Proof
We may enlarge so that contains all places over 2 and 3 and also is a PID. Then for every , we have a minimal Weierstrass equation If further has good reduction outside , we know the discriminant . Suppose we have an infinite sequence of elliptic curves having good reduction outside . Since the -unit group is finitely generated, we know that is a finite group. So we can find an infinite subsequence (still denoted by ), such that are in the same class of . In other words, for a fixed . From , we know that has -solutions . Therefore by Siegel's theorem, there are only finitely many possibilities for and . Moreover, each of the identities , gives a -isomorphism via , .
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Theorem 11 (Finiteness I)
Fix an elliptic curve . Then there are only finitely many elliptic curves which are isogenous to .

Proof
Suppose and are isogenous over . Then they have the same set of places of good reduction by Neron-Ogg-Shafarevich (the induced map is an isomorphism of -modules for all 's prime to the characteristic of the residue field and the degree of the isogeny). The result then follows from Shafarevich's theorem.
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[1]Heights and elliptic curves, Arithmetic geometry (Storrs, Conn., 1984), Springer, 1986, 253--265.

[2]Abelian Varieties (v2.00), Available at www.jmilne.org/math/.

[3]The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics), Springer, 2010.

[4]Heights in Diophantine Geometry (New Mathematical Monographs), Cambridge University Press, 2007.

[5]Diophantine Geometry: An Introduction (Graduate Texts in Mathematics), Springer, 2000.