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

The goal of this article was to study the Iwasawa theory of an abelian variety A that has complex multiplication by a complex multiplication (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 [Kâzım Büyükboduk and Antonio Lei, ‘Integral Iwasawa theory of motives for non‐ordinary primes’, 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed p ‐adic L ‐function via a reciprocity conjecture that 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 Z p ‐power extension of F for the primes failing the ordinary hypothesis of Katz.