  In this article, we study the p-ordinary Iwasawa theory of the (conjectural)
Rubin-Stark elements defined over abelian extensions of a CM field F and
develop a rank-g Euler/Kolyvagin system machinery (where 2g is the degree of
F), refining and generalizing Perrin-Riou's theory and the author's prior work.
This has several important arithmetic consequences: Using the recent results of
Hida and Hsieh on the CM main conjectures, we prove a natural extension of a
theorem of Yager for the CM field F, where we relate the Rubin-Stark elements
to the several-variable Katz p-adic L-function. Furthermore, beyond the cases
covered by Hida and Hsieh, we are able to reduce the p-ordinary CM main
conjectures to a local statement about the Rubin-Stark elements. We discuss
applications of our results in the arithmetic of CM abelian varieties.
