Martin's Blog

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.

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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.

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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.

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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.

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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.

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Dual abelian varieties over the complex numbers

Posted by martin 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.

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Polarisable complex tori are projective

Posted by martin on Saturday, 26 March 2011 at 16:07

Last time, we defined polarisations on $H_1$ Hodge structures and saw that if $A$ is a complex abelian variety, then $H_1(A)$ has a polarisation. This time we will prove the converse: if $X$ is a complex torus such that $H_1(X)$ has a polarisation, then $X$ is an abelian variety (in other words, $X$ can be embedded in projective space). The proof is based on studying invertible sheaves on $X$.

This is long, even though I have left out all the messy calculations. For full details, see Mumford’s Abelian Varieties or Birkenhake-Lange’s Complex Abelian Varieties. For the next post, you will only need to know the two statements labelled as theorems.

This theorem is a special case of the Kodaira Embedding Theorem, which tells you that any compact complex manifold is projective if it has a polarisation, but that is somewhat more difficult.

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Polarisations on Hodge structures

Posted by martin on Saturday, 26 February 2011 at 18:27

In the last post, I discussed Hodge symplectic forms. Now I shall show that the $H_1$ of an abelian variety has a polarisation, which is defined to be a Hodge symplectic form satisfying a positivity condition. The importance of polarisations is that they give a way of recognising which $H_1$ Hodge structures come from abelian varieties - I shall discuss this application next time.

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Hodge symplectic forms

Posted by martin on Saturday, 18 December 2010 at 15:00

Both the Hodge structure and the Tate module of an abelian variety come with symplectic forms which are (almost) preserved by the action of the relevant group (Mumford-Tate or Galois group). The form on the Tate module, called the Weil pairing, will require some preparation. So in this post I will construct the Hodge symplectic forms (also called the Riemann forms) on the Hodge structure. Next time I will discuss some further properties of Hodge forms.

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Images of Galois representations

Posted by martin on Saturday, 27 November 2010 at 16:22

In this post, I will continue to talk about the $\ell$-adic representations attached to abelian varieties, and in particular the images $G_\ell$ of these representations. I will define algebraic groups approximating $G_\ell$, which are often more convenient to work with. I will end by stating the Mumford-Tate conjecture, linking $G_\ell$ to the Mumford-Tate group.

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