# Martin's Blog

## 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:

1. is contained in the centre of .
2. 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.

1. Tate classes From Martin's Blog

In my last post I talked about Hodge classes on abelian varieties. Today I will talk about the analogue in l-adic cohomology, called Tate classes. These are defined to be those classes on which the action of the Galois group is given by multiplying ...