# Martin Orr's Blog

## 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 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.

### Normed modules

Let be the ring of integers of a number field. We define a normed -module to be a projective -module of finite rank together with a norm for each archimedean place of .

(A norm is a function satisfying if , if , and the triangle inequality.)

Our primary example of a normed module is the canonical module of the Néron model of an abelian variety: let be an abelian variety of dimension and its Néron model. Then is a projective -module of rank 1, and we get a norm on as follows: if then we can extend uniquely to a global canonical form on . We define 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 , then we simply choose a generator of (unique up to sign) and define to be .

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 might not be principal, so there is no generator. To get around this, we choose any at all, and add on a term that measures how far is from being a generator. Specifically, Here (which is 1 or 2).

Note that by the Chinese remainder theorem, we can write the first term of as a sum over the finite places of : The expression for is independent of the choice of by the product formula for absolute values.

### Faltings heights

We define the (unstable) Faltings height of an abelian variety using the degree of the canonical module of its Néron model: The factor reduces the dependence on the base field , but does not eliminate it. Specifically, if is a normed -module, then we have However the Néron model of need not be the base change of the Néron model of , so that . 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 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 has semistable reduction over some finite extension of .

Fact 2. If 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 to be for some finite extension such that has semistable reduction. By fact 2 it does not matter which we pick, and the stable Faltings height is independent of the base field, so it is often more convenient than the unstable Faltings height.

1. The Chowla-Selberg formula From Martin's Blog

The Chowla-Selberg formula is an equation which expresses the periods of CM elliptic curves in terms of values of the gamma function at rational arguments. Colmez conjectured a generalisation of the Chowla-Selberg formula to higher-dimensional CM ...

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

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

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.

Barinder Banwait, frequent commentator on Martin's blog, has setup his own blog at bbanwait.wordpress.com. He will be discussing Mazur's Isogeny Theorem, with a view towards the Uniform Boundedness Theorem. But he may not get there. Why not check out what the fuss is about?