Ming-Lun Hsieh recently gave a proof of the CM main conjectures under the -ordinary hypothesis of Katz in a wide variety of cases. His work builds on the prior work of Hida, Hida-Tilouine and Mainardi, all of which use variants of the Eisenstein ideal method. Using Hsieh's result and refining the rank r Euler systems machinery, I will relate the (conjectural) Rubin-Stark elements to Katz' -adic -function. When the CM field in question is quadratic imaginary, this result has been proved by Yager. Should time permit, I will explain how to approach the more mysterious non-ordinary setting with our method as well as deduce applications towards the Birch and Swinnerton-Dyer conjecture for CM abelian varieties.