Martin Orr's Blog

Weil pairings: definition

Posted by Martin Orr 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 Orr 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 Orr 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 Orr 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.

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

Dual abelian varieties over the complex numbers

Posted by Martin Orr on Tuesday, 26 April 2011 at 12:35

In this post I will define dual abelian varieties over the complex numbers. The motivation is that polarisations can be interpreted as isogenies from an abelian variety to its dual. For the moment, all this is tied to Hodge structures so only works over the complex numbers, but this is the view of polarisations which will we will generalise later to other fields.

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

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