## 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.`

Barinder Banwaitsaid 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`

`?`

Barinder Banwaitsaid 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?