Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 126 / 2 / iv / Solution

For an abelian variety $X$, consider the commutator morphismIt is the identity whenever either coordinate is the identity. The Mumford rigidity lemma applied successively to the two complete connected factors makes $c$ constant everywhere; its value at $(e,e)$ is $e$. Therefore every pair of points commutes, so the group law is commutative.