 Homomorphism associated to a line bundle on an abelian variety

For a line bundle $L$ on an abelian variety $A$, setThe translated difference is algebraically trivial, so its value belongs to the dual abelian variety $\hat{A}={\mathrm{Pic}}^0(A)$ rather than merely the whole Picard group. The Theorem of the square makes this a group homomorphism. Tensoring line bundles adds their homomorphisms; an algebraically trivial bundle gives the zero map. An ample line bundle gives a finite surjective map, a polarization of an abelian variety; its kernel measures translations preserving the bundle class.