TITLE: The Dual Abelian Variety
QUESTION [9 upvotes]: Let $A$ be an abelian variety defined over an algebraically closed field, say over $\mathbb{C}$. There is a dual abelian variety $\hat{A}$, along with a Poincare line bundle $L$ on $A\times \hat{A}$. Is there any relation between $\widehat {A\times A} $ and $\hat{A}\times \hat{A}$, for instance are they isogenous. What happens when $A$ is principally polarized, can we say relate the Poincare bundles in this case.

REPLY [18 votes]: You can define $\hat{A}$ as $\underline{\mathrm{Pic}}^0(A)$. Now, over any algebraically closed field $k$, let $V$ and $W$ be proper (irreducible, reduced) varieties. Then $\underline{\mathrm{Pic}}^0(V)\times \underline{\mathrm{Pic}}^0(W)\to\underline{\mathrm{Pic}}^0(V\times W)$ is an isomorphism. Injectivity is immediate, and surjectivity follows from the theorem of the cube (see e.g. Mumford, Abelian Varieties, section 6 in chapter II).