Martin'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