Duality of abelian varieties

In this talk the dual of an abelian variety will be defined. It will be shown that this dual is an abelian variety again. Using dual morphisms, we will finally show that the dual of the dual is canonically isomorphic to the orginal abelian variety. The author would like to thank Bas Edixhoven, Julian Lyczak and the other participants to the seminar for their valuable contributions. 0 Conventions and notation The letters A and B will be used to denote an abelian variety over a base scheme, S = Spec k, the spectrum of a field k. 1 Dual abelian variety Last week, Smit discussed the Picard functor. In our case he showed that PicA/S is a separated scheme, locally of finite type over S. It represents the relative Picard functor Sch/S → Set : T 7→ Pic(AT ) / prT Pic(T ). By construction it is a group scheme, whose identity point e ∈ PicA/S(S) is given by the structure sheaf OA. Definition 1. The dual of A is A∨ := PicA/S , the connected component of PicA/S containing the identity e. Example 2. For an elliptic curve over S, i.e. an abelian variety of dimension 1, the dual is canonically isomorphic to itself, using for example Riemann-Roch. In the rest of this section, there will be an outline of a proof of the fact that A∨ is again an abelian variety.