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

Abstract

The goal of this article is to study the Iwasawa theory of an abelian variety A that has complex multiplication by a CM field F that contains the reflex field of A, which has supersingular reduction at every prime above p. To do so, we make use of the signed Coleman maps constructed in our companion article [BL14] to introduce signed Selmer groups as well as a signed p-adic L-function via a reciprocity conjecture we formulate for the (conjectural) Rubin-Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank-g Euler-Kolyvagin systems to be applied with Rubin-Stark elements and deduce the main conjecture for the maximal Zp-power extension of F for the primes failing the ordinary hypothesis of Katz.