CM period (${\Omega }_{\infty },{\Omega }_p$)

A complex CM period compares an algebraic invariant differential on a CM elliptic curve with its analytic lattice coordinate. At an ordinary prime, a $p$-adic CM period compares the same differential with the invariant differential of the multiplicative formal group: an isomorphism $\eta$ is normalized by ${\log }_{\hat{E}}\eta (T)={\Omega }_p\log (1+T)$. Matched periods make normalized critical Hecke L-function values algebraic and give precise p-adic L-function interpolation.