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The Faltings height of an abelian variety over the rationals

Posted by Martin Orr 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 \mathbb{Q}, 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 A over \mathbb{Q}, the Faltings height is the (logarithm of the) volume of A as a complex manifold with respect to a particular volume form, chosen using the \mathbb{Q}-structure of A. 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 \mathcal{A}_g. Then he shows that the Faltings height is bounded within an isogeny class. Both of these parts are difficult.

Differential forms and cotangent spaces

Let R be a ring and X a smooth R-scheme of relative dimension n. We can then define:

  1. the cotangent sheaf \Omega^1_{X/R}, a locally free sheaf of rank n;
  2. the cotangent space T_s^\vee (X/R) for each R-point s of X, a projective R-module of rank n.

The idea is that the cotangent sheaf consists of differential forms, that is things locally of the form f\,dg (or sums of such terms) where f, g are functions on X.

More precisely, let A be an R-algebra.

We define the module of differential forms \Omega^1_{A/R} to be the A-module generated by symbols df for f \in A, subject to the relations  d(f+g) = df + dg, \quad df = 0 \text{ if } f \in R, \quad d(fg) = f\,dg + g\,df.

For example, if A is the polynomial ring R[X_1, \dotsc, X_n], then \Omega^1_{A/R} is the free A-module of rank n generated by dX_1, \dotsc, dX_n.

Now on an R-scheme X, we can glue together the modules of differential forms on affine open subschemes to get a sheaf of \mathcal{O}_X-modules \Omega^1_{X/R}, which is the cotangent sheaf of X.

If s : \mathrm{Spec} R \to X is an R-point of X, then we can pull back the cotangent sheaf to get s^* \Omega^1_{X/R}, a sheaf on \mathrm{Spec} R. We define the cotangent space T_s^\vee (X/R) to be the global sections of this sheaf on \mathrm{Spec} R.

Thus far we have defined the cotangent space and cotangent sheaves for any scheme. If we suppose that X \to R is smooth, then it turns out that \Omega^1_{X/R} is locally free. It follows that s^* \Omega^1_{X/R} is also locally free, and so T_s^\vee(X/R) is a projective R-module.

The canonical bundle and volume forms

Again let X be a smooth R-scheme of relative dimension n. Since \Omega^1_{X/R} is locally free of rank n, its n-th exterior power is a line bundle on X. We call this the canonical bundle K_{X/R} = \bigwedge^n \Omega^1_{X/R}.

The central object in the definition of the Faltings height is the pullback of the canonical bundle at an R-point, s^* K_{X/R} (or equivalently \bigwedge^n T_s^\vee (X/R)). I don't think that this has a standard name, but it seems reasonable to call it the canonical module of X/R at s. It is a projective R-module of rank 1.

Let us work for a moment over the complex numbers. Canonical forms on X are related to volume forms on the complex manifold X(\mathbb{C}).

A volume form on an oriented real manifold M is something which you can integrate over M to calculate its volume. Its degree (i.e. the number of dxs which appear in the volume form) must be equal to \dim M.

Now X(\mathbb{C}) has dimension 2n as a real manifold, so a volume form has degree 2n. Given a global section \omega of K_{X/\mathbb{C}}, we can get a volume form by taking \omega \wedge \bar\omega. We define the volume of X with respect to \omega to be  \mathrm{Vol}(X, \omega) = \left( \frac{i}{2} \right)^n \int_{X(\mathbb{C})} \omega \wedge \bar\omega. The factor i^n ensures that the result is a positive real number and the factor of 2^{-n} is a convention. (This convention comes from the fact that on the complex plane with coordinate z = x+iy, the natural real volume form dx \wedge dy is equal to \frac{i}{2} dz \wedge d\bar{z}.)

The Faltings height

Let A be an abelian variety of dimension g over \mathbb{Q}. The Faltings height of A is defined to be the volume of A(\mathbb{C}) with respect to a particular choice of canonical form.

Let \mathcal{A} be the Néron model of A. All you need to know about the Néron model is that it is a smooth group schemes over \mathbb{Z} such that \mathcal{A} \times_\mathbb{Z} \mathbb{Q} = A and that every abelian variety has a unique Néron model.

Let e : \mathrm{Spec} \mathbb{Z} \to \mathcal{A} be the zero section. By what we said above, the canonical module \bigwedge^g T_e^\vee (\mathcal{A}/\mathbb{Z}) is a projective \mathbb{Z}-module of rank 1, and hence is isomorphic to \mathbb{Z}. So let \tilde\omega be a generator of this module, unique up to sign. (This is where we have a problem over other fields, as the module need not be principal, and even if it is the generator is not unique.)

We use the following theorem, which uses both the group structure and the properness of an abelian variety.

Theorem. Let A be an abelian variety of dimension g over a field k. Then the restriction maps \Gamma(A, \Omega^1_{A/k}) \to T_0^\vee (A/k) and \Gamma(A, K_{A/k}) \to \bigwedge^g T_0^\vee(A/k) are isomorphisms.

We can restrict \tilde\omega to the generic fibre of \mathcal{A}, giving an element \omega_0 \in \bigwedge^n T_0^\vee (A/\mathbb{Q}). By the theorem, we can extend \omega_0 to a global section \omega of the canonical bundle K_{A/\mathbb{Q}}. Finally the Faltings height is defined to be  h(A) = -\tfrac{1}{2} \log \mathrm{Vol}(A, \omega). I think we need the \frac{1}{2} here to agree with the definition for a general number field in Faltings' paper, but it is of very little importance - a variety of normalisations are used. The minus sign is needed so that heights are bounded below by an absolute constant.

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

Trackbacks

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Comments

  1. Barinder Banwait said on Sunday, 20 November 2011 at 11:36 :

    Question 1. Let A,A' be two abelian varieties over \mathbb{Q} which are not isomorphic over \mathbb{Q}. Are their Faltings heights different? If "No, not necessarily", what is a good example? If "Yes", then what would a big difference in their heights mean? Would it measure how "different" they are? Would a big difference correspond to some other big arithmetic difference?

    Remark 2. Concerning my question 1, I have asked myself questions in a similar vein in the past about what it means for a number field to have a "big" class number, and trying to understand precisely what "the class number measures the failure of unique factorisation" means.

    Question 3. Is the "classical height" bounded within an isogeny class? I'd guess yes, but it is very hard, perhaps impossible, to prove directly (that is, without introducing the Faltings height). For otherwise, if it were easy to prove directly, there would have been no need for Faltings to introduce Faltings height to prove Finiteness I.

    Problem 4. I wonder about any link between the Faltings height of A/\mathbb{Q} and the image of the l-adic Galois representations \rho_{A,l} of A. It is known (I think) that for l > c(A,\mathbb{Q}) prime, \rho_{A,l} is surjective. Can this bound be made explicit in terms of the Faltings height? More generally (maybe), if I'm looking for (or trying to disprove the existence of) an abelian variety of dimension 2 with an unusual property in its 13-adic representation, can I say "Oh, its Faltings height must be at most 92" or something?

    Question 5. What values in \mathbb{R}_{\geq 0} can the Faltings height take? Given any r \in \mathbb{R}_{\geq 0}, can one construct an A with Faltings height precisely r?

    Question 6. I've seen Néron models before, mainly in Mazur's work in the 70s which I once tried to understand. But their role here makes me uneasy. The volume form comes from a global section of the canonical bundle attached to the variety-viewed-over-\mathbb{C}. Why should one need to work with the Néron model, an altogether more complicated object? Put another way, why should I care about the other fibres in \mathcal{A}, or the reductions of A?

    Request 7. Please forgive my pedantries.

    Pedantry 8. Every instance of "Faltings' proof" and "Faltings' paper" in your post should be replaced with "Faltings's proof" and "Faltings's paper", respectively. "Faltings' proof" suggests that it is a proof due to multiple persons each named Falting. Note however that it is indeed "The Faltings Height" and not "The Falting's height"; Milne would refer to this latter one as a "truly gruesome error".

    Further Pedantries 9. Last line of paragraph 1. And is "canonical module of X/R at s" really the best name? I'd replace "module" with "bundle".

    Hope 10. I look forward to seeing the two hard steps as mentioned in paragraph 3.

  2. Barinder Banwait said on Tuesday, 22 November 2011 at 12:39 :

    Regarding my problem 4, I've just found some interesting results for elliptic curves in the recent preprint of Bilu, Parent and Rebolledo, for example Theorem 2.2; let p \geq 3 be prime, r any positive integer, K a quadratic number field, and P a point on Y_0(p^r)(\mathcal{O}_K). Then the Faltings height h(P) of the elliptic curve corresponding to P is bounded above by 2p\log p + 4p. There are other such results regarding isogenies in that paper.

  3. Martin Orr said on Thursday, 24 November 2011 at 14:23 :

    I know very little about the Faltings height beyond what is in this post. I am afraid I do not plan to realise your Hope 10, though I will say something about the Masser-Wüstholz proof of Finiteness I which gives a bound on the minimum degree of an isogeny between two abelian varieties. The Faltings height appears in this bound, but I believe that its appearance there is entirely different from its appearance in Faltings' proof.

    Question 1. Faltings shows that, if two abelian varieties A and A' are isogenous with an isogeny satisfying certain conditions, then h(A) = h(A'). Hence over some number field there are non-isomorphic isogenous abelian varieties with the same Faltings height. I do not know if such a pair exists over \mathbb{Q}.

    Do there exist non-isogeous abelian varieties with the same Faltings height? And more generally, if A and A' are not isogenous, then does the difference between their heights mean anything? I have no idea.

    Question 3. Yes, the classical height is bounded within an isogeny class, as a consequence of the boundedness of the Faltings height.

    Problem 4. There are links between the Faltings height and isogenies, and hence l-adic representations. Masser and Wüstholz proved (Bulletin of the LMS 25, 1993):

    Theorem. There are absolute constants c and \gamma such that if E is an elliptic curve over a number field K without CM, then the image of the Galois representation contains \mathrm{SL}_2(\mathbb{F}_\ell) for all  \ell > c (\max([K:\mathbb{Q}], h(E))^\gamma (and so the image is equal to \mathrm{GL}_2(\mathbb{F}_\ell) if also \ell does not divide the discriminant).

    Question 5. There are only countably many abelian varieties over \mathbb{Q} so the Faltings height cannot take all values in \mathbb{R}_{\geq 0}.

    Furthermore it seems (http://mathoverflow.net/questions/72829/which-curves-have-stable-faltings-height-greater-or-equal-to-1) that it is not true that the Faltings height is always positive. However there are a variety of different normalisations used so maybe for some normalisations it is always positive.

    Question 6. The global sections of the canonical bundle over \mathbb{C} form a 1-dimensional \mathbb{C}-vector space. So if we want to get a number, we need some way of choosing an element of this space. By looking at the canonical bundle over \mathbb{Q}, we still get a \mathbb{Q}-vector space, which has no distinguished element. By taking a model over \mathbb{Z}, we get a \mathbb{Z}-module of rank 1, which does now have a unique generator (up to sign). But then we need some way of choosing a model over \mathbb{Z}, and the Néron model is the obvious thing to choose.

    Less naïvely, the height of a point in projective space is a sum of terms for each place of \mathbb{Q}. So if we want a notion of "height" of an abelian variety, then yes we should expect to see contributions from the reductions. Over \mathbb{Q} it is possible to hide everything except the archimedean contribution, which made this post simpler but more mysterious Next time this should be more apparent with general number fields.

    Further Pedantries 9. It seems to me confusing to have both "canonical bundle of X/R" (a line bundle on X) and "canonical bundle of X/R at s" (an R-module). This is the equivalent of the "cotangent space", which is the specialisation of the "cotangent sheaf" at a point, so "canonical space" would be a reasonable name but seemed ugly to me.

  4. Matthew Emerton said on Friday, 02 December 2011 at 01:00 :

    Regarding question 1, it seems that you are asking whether there are isomorphic but non-isogenous elliptic curves over Q. The answer is yes; you can look at Cremona's tables and he tells you the size of each isogeny class, and already for lowest possible conductor, namely 11, there are three non-isomorphic members of the isogeny class, namely X_0(11), X_1(11), and a third curve.

  5. Barinder Banwait said on Wednesday, 07 December 2011 at 00:33 :

    Thank you Matthew for your comment. In the case of elliptic curves over Q, my question becomes: Let E_1, E_2 be two non-isomorphic elliptic curves over Q. Must their Faltings heights be different?

    Probably no. That is, I'd expect there to exist a pair of non-isomorphic elliptic curves with the same Faltings height.

    But I'm not sure that isogenous curves have the same Faltings height. For E_1 = X_0(11) and E_2 = X_1(11), using SAGE one finds the 'heights' to be 18.619... and 38.507.... But this may very well be a different notion of height.

    Martin's question of whether there is a pair of non-isogenous abelian varieties over Q with the same Faltings height is a very interesting one.

    I guess I'm really confused with this: to what extent does the Faltings height 'determine' an abelian variety over a number field K?

    (Here 'determine' can mean 'up to isomorphism over K', 'up to isogeny over K', or 'up to isomorphism/isogeny over \bar{K}'.)

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