Arithmetic-Geometric Means of complex numbers and evaluation of modular functions

We describe the deep connection, already studied by Gauss, between the arithmetic-geometric mean (AGM) of complex numbers and modular forms known as theta constants. We then show how this can be used to devise an algorithm for the numerical evaluation of modular functions (such as the elliptic function j) with a quasi-optimal time complexity. As a byproduct, we also obtain an algorithm for the numerical evaluation of complex logarithms, again with quasi-optimal time complexity.