Homomorphism associated to a line bundle on an abelian variety (${\varphi }_L$)

For a line bundle $L$ on an abelian variety $X$,is a homomorphism by the Theorem of the square.

For an abelian variety,is the subgroup of translation-invariant line bundles.

The seesaw theorem determines a line bundle on a product from its restrictions to fibers: if it is trivial on all fibers over one factor and on one transverse section, then it is trivial.