Martin's Blog : Older posts, page 2

Weil pairings: the skew-symmetric pairing

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

Last time, we defined a pairing $e_\ell : T_\ell A \times T_\ell (A^\vee) \to \lim_\leftarrow \mu_{\ell^n}.$ By composing this with a polarisation, we get a pairing of $T_\ell A$ 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 cyclotomic character, as I promised a long time ago. This tells us that the image of the $\ell$ -adic Galois representation of $A$ is contained in $\operatorname{GSp}_{2g}(\mathbb{Q}_\ell)$ . This is the end of my series on Mumford-Tate groups and $\ell$ -adic representations attached to abelian varieties.

no comments Tags abelian-varieties, alg-geom, hodge, maths, number-theory Read more...

Weil pairings: definition

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

Recall that for an abelian variety $A$ over the complex numbers, $H_1(A^\vee, \mathbb{Z})$ is dual to $H_1(A, \mathbb{Z})$ (this is built in to the analytic definition of $A^\vee$ ). Since $T_\ell A \cong H_1(A, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{Z}_\ell$ , this tells us that $T_\ell(A^\vee)$ is dual to $T_\ell A$ (as $\mathbb{Z}_\ell$ -modules). We would like to show that this is true over other fields as well, which we will do by constructing the Weil pairings.

no comments Tags abelian-varieties, alg-geom, hodge, maths, number-theory Read more...

Dual varieties over general fields

Posted by martin on Friday, 24 June 2011 at 17:26

Today we will construct dual abelian varieties over number fields. We use the universal property from two posts ago to define dual abelian varieties, then we give a simple construction inspired by the complex case. Proving that this construction satisfies the universal property is harder; in the case of number fields, we will use Galois descent to deduce it from the complex case which we already know analytically.

2 comments Tags abelian-varieties, alg-geom, hodge, maths Read more...

Line bundles and morphisms to the dual variety

Posted by martin on Saturday, 28 May 2011 at 15:25

Over the complex numbers, the dual of an abelian variety $A$ is defined to have a Hodge structure dual to that of $A$ . Hence morphisms $A \to A^\vee$ can be interpreted as bilinear forms on the Hodge structure of $A$ . Of particular importance are the morphisms corresponding to Hodge symplectic forms.

Last time we saw that $A^\vee$ can also be interpreted as a group of line bundles on $A$ . Today we will use this interpretation to define morphisms $\phi_\mathcal{L} : A \to A^\vee$ which turn out to be the same as those corresponding to Hodge symplectic forms. Then we generalise the definition of $\phi_\mathcal{L}$ to base fields other than $\mathbb{C}$ , which we will use next time in constructing dual abelian varieties over number fields.

2 comments Tags abelian-varieties, alg-geom, hodge, maths Read more...

Dual abelian varieties and line bundles

Posted by martin on Monday, 09 May 2011 at 14:30

The definition I gave last time of dual abelian varieties was very much dependent on complex analytic methods. In this post I will explain how dual varieties can be interpreted geometrically: the points of $A^\vee$ correspond to a certain group of line bundles on $A$ . We construct a single line bundle $\mathcal{P}$ on the product $A \times A^\vee$ , the Poincaré bundle, such that all line bundles on $A$ arise as restrictions of $\mathcal{P}$ , and show that the pair $(A^\vee, \mathcal{P})$ satisfies a universal property.