The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic-Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over $\C$, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over $\C$. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both \Magma\ and \Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.