Coates–Wiles homomorphism (${\delta }_k(u)={\frac{d^k}{d{z}^k}\log f_u({\exp }_{\hat{E}}z)|}_{z=0}$)

The $k$th logarithmic derivative in the additive formal logarithm coordinate defines the Coates–Wiles homomorphism. Products of local units become sums under this map. For elliptic units, differentiating the underlying complex elliptic function expresses the result in terms of critical Hecke L-function values divided by a CM period.