[math-fun] Unitary Transforms of Elliptic Curves?

The proof that canonical or symplectic transformations leave invariant the real period of an elliptic curve is not too difficult using Jacobian matrices of determinant 1. Generalization of the proof seems to imply the corollary that complex determinant transformations from unitary act as rotation of the period parallelogram through the complex plane. It would be nice to have an intelligible reference on the full theorem, if it is correct. A few years ago I skimmed through every “Yellow Peril” textbook I could find on the subject (including Silverman), and didn’t find the answers I was looking for. Did I miss something, or am I just worried about another fact that is “obviously” right (or wrong)? — Brad