Let $E$ be a CM elliptic curve over the rationals and $p$ an odd prime. If the $p^{\infty}$-Selmer group of $E$ has $\mathbf{Z}_{p}$-corank one, we show that the analytic rank of $E$ is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy in the ordinary case.