Main conjectures for CM fields and a Yager-type theorem for Rubin-Stark elements

Abstract

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 Graphic), 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.