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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), 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.

Main Conjectures for CM Fields and a Yager-Type Theorem for Rubin–Stark Elements