Theorem of the cube

Remarks

The standard proof uses the seesaw principle, which detects triviality of a line bundle on a product from its restrictions to fibres. It combines this with a reduction to the case where $Z$ is a curve.

Applied to an abelian variety $A$, the theorem gives a formula on $A \times A \times A$. Let $p_i$ be the projections and $p_{ij} = p_i + p_j$, $p_{123} = p_1 + p_2 + p_3$ the sum maps to $A$. For every line bundle $L$ on $A$, the line bundle $$p_{123}^{*} L \otimes p_{12}^{*} L^{-1} \otimes p_{13}^{*} L^{-1} \otimes p_{23}^{*} L^{-1} \otimes p_1^{*} L \otimes p_2^{*} L \otimes p_3^{*} L$$ is trivial. Let $t_x$ be translation by a point $x$ of $A$. Pulling back along maps into $A \times A \times A$ gives the theorem of the square: $t_{x+y}^{*} L \otimes L \cong t_x^{*} L \otimes t_y^{*} L$ for all points $x, y$. It also gives $[n]^{*} L \cong L^{(n^2 + n)/2} \otimes [-1]^{*} L^{(n^2 - n)/2}$ for multiplication by $n$. So for a symmetric line bundle, with $[-1]^{*} L \cong L$, one gets $[n]^{*} L \cong L^{n^2}$.

Going deeper

For a student

The theorem of the cube is a basic tool in the algebraic theory of abelian varieties. The theorem of the square that follows from it gives a homomorphism $A \to \operatorname{Pic}(A)$, $x \mapsto t_x^{*} L \otimes L^{-1}$, which lands in the points of the dual abelian variety. For an ample $L$ this map is an isogeny, the polarization attached to $L$. The formula $[n]^{*} L \cong L^{n^2}$ for symmetric $L$ is a key step in computing the degree of multiplication by $n$, and in the theory of heights.