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 2g = [F : Q]), 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.

Coleman-adapted Rubin–Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties

“…In this Appendix we recall a result that the first author proved in [4] which shows the existence of Kolyvagin systems for the Selmer structure F L on T. Even though these Kolyvagin systems do exist unconditionally, they are related (via Theorem A.11) to the L-restricted Kolyvagin system κ Ξ obtained from the (conjectural) Perrin-Riou-Stark elements and that we utilized above.…”

“…The proof of Theorem B.3 is identical to that of [4,Theorem A.14] and in what follows, we only indicate the key points in the argument and state some of the technical consequences which we also need in the main body of this article. Forᾱ = (α 1 , .…”

“…We fix forever a prime p > 3 that is unramified in F . Iwasawa theory over the CM field F has been studied in [4,[7][8][9]12] under a certain p-ordinary hypothesis of Katz. Our goal here was to carry out a similar task in the absence of this hypothesis; as a matter of fact, in some sense in the other extreme case when: (H.K.)…”

On the Iwasawa theory of CM fields for supersingular primes

“…To obtain the generalization above we make use of the Rubin-Stark elements. To do so, the CM rank-g Euler/Kolyvagin system machinery developed by the author in [Büy14] (relying crucially on the p-ordinary hypothesis (1.1)) requires a non-trivial refinement. This is one of the major tasks we carry out in this article.…”

“…As further explained in [Büy14, §3.1], the Rubin-Stark elements may be used to construct an Euler sytem of rank g for T (in the sense of [PR98], as appropriately generalized in [Büy10] so as to allow denominators). We omit the details here and refer the reader to [Büy14]. This Euler system of rank g is a collection C (g)…”