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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 Zp-power extension of F for the primes failing the ordinary hypothesis of Katz. [ABSTRACT FROM AUTHOR]

We show that any abelian surface with multiplication by the quaternion ℚ-algebra of discriminant 6, with field of moduli ℚ and which is a Jacobian in characteristic 2 and 3, has infinitely many primes of superspecial reduction. This is done by examining complex multiplication points in characteristic 0 and p and the values of a certain j-function on the associated moduli space at these points. [ABSTRACT FROM AUTHOR]