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

TL;DR

This paper advances the understanding of Iwasawa theory for CM fields by developing a new Euler system machinery, proving a Yager-type theorem, and connecting Rubin-Stark elements with p-adic L-functions, with implications for CM abelian varieties.

Contribution

It introduces a refined Euler/Kolyvagin system framework for Rubin-Stark elements, extending Perrin-Riou's theory and proving a new Yager-type theorem for CM fields.

Findings

Proved a Yager-type theorem relating Rubin-Stark elements to p-adic L-functions.

Reduced CM main conjectures to local statements about Rubin-Stark elements.

Extended the theory of Euler systems in the context of CM fields.

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 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.