 Polarization of an abelian variety

Over an algebraically closed field $k$, an ample line bundle $L$ on an abelian variety $A$ defines a homomorphismwhere $A^{\vee }$ is the dual abelian variety and $t_x$ is translation by $x$. This isogeny is a polarization of an abelian variety. Over a general field, a polarization is an isogeny that has this form after extending to an algebraic closure; an inducing ample line bundle need not descend to the original field.^[ref]

A principal polarization is an isomorphism $A\to A^{\vee }$. An elliptic curve with its degree-one origin divisor gives a basic example. A polarization supplies positivity data relating the geometry of line bundles to the algebra of isogenies; twisting $L$ by a degree-zero line bundle does not change ${\lambda }_L$.