Maths > Abelian varieties > Polarisations, dual abelian varieties and the Weil pairing

## Weil pairings: the skew-symmetric pairing

Posted by Martin Orr on Tuesday, 06 September 2011 at 13:52

Last time, we defined a pairing
```
By composing this with a polarisation, we get a pairing of
```

```
with itself.
This pairing is symplectic; the proof of this will occupy most of the post.
```

We will also see that the action of the Galois group on this pairing is given by the (inverse of the) cyclotomic character,
as I promised a long time ago (in the comments).
This tells us that the image of the `-adic Galois representation of `

` is contained in `

```
.
This is the end of my series on Mumford-Tate groups and
```

`-adic representations attached to abelian varieties.`

### Definition of the pairing

Let ```
be a polarisation.
We define
```

```
by
```

```
where
```

` is the Weil pairing defined last time.`

Taking the inverse limit over powers of `, we get the `

```
-adic pairing
```

### The theta group

We would like to show that the pairings ```
are symplectic.
The elegant way to do this is by introducing the theta group of a line bundle.
```

Let ` be a line bundle on `

```
.
We define the
```

*theta group* of ```
to be
```

```
This is a group under the operation
```

```
The inverse is given by
```

```
(The formula for
```

` is functorial in `

`, so really this is not just a group but the functor of points of a group scheme.)`

Projection onto the first factor gives a surjection ```
where
```

```
.
We are mainly interested in
```

*non-degenerate line bundles* i.e. those for which ```
is finite;
we have already proved that ample line bundles are non-degenerate.
```

The kernel of ` is the group of automorphisms of `

```
, which is the scalars.
So we have a short exact sequence of group schemes
```

This short exact sequence has the following two properties:

`is contained in the centre of`

`.`

`is commutative.`

These two properties imply that the commutator ` of any `

` lies in `

`, and that `

` depends only on `

` and `

```
.
So the commutator induces a function
```

```
,
and a calculation shows that this is a bihomomorphism.
```

An explicit calculation of the commutator gives
```
where
```

` and `

` are in `

`.`

The commutator pairing is obviously symplectic.
So in order to show that ` is symplectic, it will suffice to relate it to `

`.`

Theorem.Let`be a polarisation. If`

`and`

`such that`

`, then`

Proof.We must first check that`so that`

`is defined. Since`

`, we have`

`So`

`. Likewise`

`so that`

`.`

Choose an isomorphism

`. Thanks to the canonical isomorphisms`

`we can also interpret`

`as an isomorphism`

`. Hence, by the definition of the Weil pairing, we have`

Since the line bundles

`and`

`are not just isomorphic but equal, we have`

`.`

Substituting

`and`

`in the commutator formula gives`

### The Galois action

All the pairings we have defined are compatible with the Galois action in that if ```
, then
```

Now ` is a free `

```
-module of rank 1,
traditionally written as
```

```
.
The Galois action on this module is given by a character
```

` called the `

*-adic cyclotomic character*.
Concretely

Meanwhile the natural action of ` on the space `

` of symplectic forms on `

```
is given by
```

```
So we see that
```

The representation of ` on `

` preserves the non-degenerate symplectic pairing `

```
up to a scalar,
so the image of this representation is contained in
```

```
.
We saw long ago that the Mumford-Tate group is contained in
```

```
,
so this was expected.
```