Optimised addition sequences for eta and theta functions

The main ingredient of complex multiplication algorithms for elliptic curves that compute class and modular polynomials via floating point approximations is the evaluation of Dedekind's η- and of more general ϑ-functions. While algorithms are known that are asymptotically quasi-linear in the desired precision, in practice it is usually faster to evaluate lacunary power series. It has been observed experimentally that particularly short addition sequences exist for the specially structured exponents of η and ϑ. A leisurely stroll through classic number theory will provide us with proofs of this fact.