## 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.`