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

## Weil pairings: definition

Posted by Martin Orr on Monday, 29 August 2011 at 17:27

Recall that for an abelian variety ```
over the complex numbers,
```

` is dual to `

```
(this is built in to the analytic definition of
```

```
).
Since
```

```
,
this tells us that
```

` is dual to `

` (as `

```
-modules).
We would like to show that this is true over other fields as well,
which we will do by constructing the Weil pairings.
```

### The pairing over

Let us first unpack the duality between ` and `

` over the complex numbers.`

Recall that the isomorphism ```
is
obtained as the inverse limit of isomorphisms
```

```
where
```

```
.
So the pairing
```

```
is obtained as the inverse limit of pairings
```

The finite-level pairings ` (where `

` above, but actually could be any integer) are defined as follows:`

Let ` and `

` be the analytic universal covering maps.`

Suppose that ` and `

```
.
Choose
```

` and `

` such that `

` and `

`.`

Under the isomorphism ```
,
```

` corresponds to `

` and `

` to `

```
so we have
```

```
Because
```

` and `

` are `

`-torsion, this is an integer and independent (mod `

`) of the choice of `

` and `

`.`

### The pairing over

in terms of line bundles

We will now describe this pairing in terms of line bundles, still over the complex numbers.

When we work over other fields, it will be more natural to take the pairing as mapping into the group ` of `

`-th roots of unity, rather than `

```
.
Over
```

`, we identify these groups via `

```
.
So we replace the pairing
```

```
by
```

Recall that under the analytic construction of the isomorphism ```
,
```

` corresponds to the line bundle `

```
The sections of
```

` over an open set `

` are holomorphic functions `

```
satisfying
```

So ` is the factor by which sections of `

` are multiplied when we translate them by `

`.`

This "translation by ```
" happens in the universal cover, so cannot be generalised directly to other fields.
But really everything happens within the finite cover
```

`, or equivalently the finite cover `

`.`

Translating ` by `

` corresponds to translating `

` by `

```
,
so informally
```

` is the factor by which sections of `

` are multiplied when we translate them by `

`.`

### General definition of the Weil pairing

We now define a pairing ` for an abelian variety over an arbitrary field `

`, for `

` an integer not divisible by `

`.`

Let ` be an abelian variety over an arbitrary field `

` and let `

`, `

```
.
Let
```

` be a line bundle on `

` corresponding to `

` under the isomorphism `

`.`

The idea, as in the last paragraph on the complex case, is to look at what happens when we translate ` by `

```
.
Since
```

`, we have `

` and so `

```
.
This is a genuine equality, not just an isomorphism of line bundles.
```

However, we no longer have a concrete description of sections of ```
as in the complex case.
So instead of looking at what happens to sections when we translate, we will look at what happens to isomorphisms
```

```
.
Of course, before doing this, we need to know that such an isomorphism exists.
It does, using
```

` and applying the following theorem which can be proved by the Theorem of the Cube.`

Theorem.Let`be a line bundle on`

`homologically equivalent to`

`. Then`

`.`

Choose an isomorphism ```
.
We can translate this to get an isomorphism
```

```
.
By the equalities
```

` and `

```
,
we can view
```

` as an isomorphism `

`.`

Now the composition
```
is an automorphism of
```

```
,
but the only automorphisms of a line bundle on a complete variety are scalars.
```

So we define ` to be the scalar `

```
.
A little calculation shows that this defines a bilinear pairing
```

```
,
and we deduce that the image is in
```

` because `

` is `

`-torsion.`

Another calculation shows that
```
and so we can take the inverse limit of the pairings
```

```
to get a pairing
```

Any of ` or `

` are called `

*Weil pairings*.

### Remarks

If you check carefully for `, you will find that the pairing `

```
defined in the previous section is the reciprocal of the one defined analytically.
This disagreement of signs seems to be the standard (anti-)convention.
```

The pairing ` gives a map `

```
.
In order to show that these modules are isomorphic, we need to show that the pairing is nondegenerate.
This can be done observing that the correspondence
```

```
specialises to
```

```
and that
```

`-actions on `

` biject with characters `

`.`

In the above construction, we could replace ` by any separable isogeny `

```
.
The construction then shows that
```

` is dual to `

`.`