= Cohomological proof of the cube theorem for elliptic curves
If a <line bundle> on $E^3$ is trivial on each zero-coordinate face, its <First Chern class> vanishes: the integral second cohomology consists of single-factor second cohomology and cross-products of first cohomology from pairs, and each summand is detected on a coordinate face. The <holomorphic exponential sequence> then puts the bundle in $\operatorname{Pic}^0(E^3)$. The decomposition of first cohomology, including <Dolbeault cohomology>, identifies this group with $\operatorname{Pic}^0(E)^3$. Triviality on the three coordinate axes makes all three factors trivial. Applying this cube principle to the alternating partial-sum pullbacks proves the usual addition-map <Theorem of the Cube>.
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