In the (very nice) book "Diophantine geometry" by Hindry and Silverman there is a proof of the Appell-Humbert theorem in Exercise A.5.5. I believe that it contains a serious mistake and I want to find out if I got it right.

The statement of the Appell-Humbert theorem [e.g. https://en.wikipedia.org/wiki/Appell%E2%80%93Humbert_theorem ] is that there is a bijection between line bundles on a complex torus and pairs $(H, \alpha)$ where $H$ is a certain Hermitian form and $\alpha$ is a semicharacter on the lattice. In the proof given in the book such a pair is constructed for a divisor using the corresponding theta function. The catch is, on a general complex torus not every line bundle comes from a divisor (an example may be found in [J.D.Lewis, A survey of the Hodge Conjecture, Lecture 5]).