Why Line Bundles?

Theorem. (Liouville) Every bounded entire function \(f:\mathbb{C}\to\mathbb{C}\) is constant.

Since an elliptic curve \(E=\mathbb{C}/\Lambda\) is compact, a holomorphic function \(f:E\to\mathbb{C}\) pulls back to a bounded entire function on $\mathbb{C}$.

By Liouville, it must be constant.

Therefore every holomorphic function on $E$ is constant. Ordinary holomorphic functions are not rich enough to construct a projective embedding.

To get more holomorphic objects, we replace functions by sections of line bundles: \(s\in H^0(E,L).\)

These sections behave like holomorphic functions locally, but can transform nontrivially between charts.