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

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.