 Theorem of the square

For a line bundle $L$ on an abelian variety $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]