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The Faltings height and normed modules

Posted by Martin Orr 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 \mathbb{Q} only, and we used two properties of \mathbb{Q}: 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.

Normed modules

Let \mathcal{O}_K be the ring of integers of a number field. We define a normed \mathcal{O}_K-module to be a projective \mathcal{O}_K-module M of finite rank together with a norm \lVert \cdot \rVert_v : M \otimes_{\mathcal{O}_K} K_v \to \mathbb{R} for each archimedean place v of K.

(A norm is a function \lVert \cdot \rVert_v : M \otimes_{\mathcal{O}_K} K_v \to \mathbb{R} satisfying \lVert x \rVert_v > 0 if x \neq 0, \lVert \lambda x \rVert_v = \lvert \lambda \rvert_v \lVert x \rVert_v if \lambda \in K_v, and the triangle inequality.)

Our primary example of a normed module is the canonical module of the Néron model of an abelian variety: let A/K be an abelian variety of dimension g and \mathcal{A}/\mathcal{O}_K its Néron model. Then M = \bigwedge^g T_e^\vee (\mathcal{A}/\mathcal{O}_K) is a projective \mathcal{O}_K-module of rank 1, and we get a norm on M \otimes K_v as follows: if \omega_0 \in M \otimes K_v then we can extend \omega_0 uniquely to a global canonical form \omega on A/K_v. We define  \lVert \omega_0 \rVert_v = \sqrt{\mathop{\mathrm{Vol}}(A(\bar{K_v}), \omega)}.

We now define the degree of a normed module of rank 1, which gives us the Faltings height of an abelian variety. (It seems a bit odd to me to call this a degree, because I expect a degree to be an integer, but I believe that the justification comes from Arakelov intersection theory.)

If \mathcal{O}_K = \mathbb{Z}, then we simply choose a generator m of M (unique up to sign) and define \deg M to be -\log \lVert m \rVert_\infty.

Over other number fields, there are two problems with this definition. Firstly there may be more than one archimedean place, so which one do we choose? We simply take a sum over all of them.

More seriously, the module M might not be principal, so there is no generator. To get around this, we choose any m \in M at all, and add on a term that measures how far m is from being a generator. Specifically,  \deg M = \log \#(M/\mathcal{O}_K.m) - \sum_{v \mid \infty} \epsilon_v \log \lVert m \rVert_v. Here \epsilon_v = [K_v : \mathbb{R}] (which is 1 or 2).

Note that by the Chinese remainder theorem, we can write the first term of \deg M as a sum over the finite places of K:  \log \#(M/\mathcal{O}_K.m) = \sum \log \#((M \otimes_{\mathcal{O}_K} \mathcal{O}_v) / \mathcal{O}_v.m). The expression for \deg M is independent of the choice of m \in M by the product formula for absolute values.

Faltings heights

We define the (unstable) Faltings height of an abelian variety A/K using the degree of the canonical module of its Néron model:  h(A/K) = \frac{1}{[K:\mathbb{Q}]} \deg \bigwedge^g T_e^\vee (\mathcal{A}/\mathcal{O}_K).

The factor 1/[K:\mathbb{Q}] reduces the dependence on the base field K, but does not eliminate it. Specifically, if M is a normed \mathcal{O}_K-module, then we have  \deg (M \otimes_{\mathcal{O}_K} \mathcal{O}_L) = [L:K] \deg M.

However the Néron model of A_L need not be the base change of the Néron model of A, so that h(A_L) \neq h(A). For example an elliptic curve with additive reduction may have good reduction after a finite extension of the base field.

We say that an abelian variety A has semistable reduction if every reduction of its Néron model is a semiabelian variety, that is an extension of an abelian variety by a torus. In the case of an elliptic curve, this is saying that it has good or multiplicative reduction everywhere.

We have the following two facts:

Fact 1. Any abelian variety A/K has semistable reduction over some finite extension L of K.

Fact 2. If A has semistable reduction, then the identity component of its Néron model is unchanged by finite extensions of the base field.

So we define the stable Faltings height h_F(A) to be h(A_L) for some finite extension L/K such that A_L has semistable reduction. By fact 2 it does not matter which L we pick, and the stable Faltings height is independent of the base field, so it is often more convenient than the unstable Faltings height.

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

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Comments

  1. Barinder Banwait said on Friday, 03 February 2012 at 20:54 :

    Let C/K be a curve over a number field, and J its Jacobian variety. There are various notions of the height of C, and I'll let you choose your favourite one. Is there any link between the height of C, and the Faltings height of J?

  2. Barinder Banwait said on Saturday, 04 February 2012 at 02:00 :

    Allow me a moment of shameless self-advertising to the readership of Martin's blog.

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