In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33-62], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mem. Amer. Math. Soc. 168 (2004), no. 799] and prove a Gras-type conjecture, relating these Kolyvagin systems to appropriate ideal class groups, refining the results of Rubin [J. Reine Angew. Math. 425 (1992), 141-154].Comment: 27 pages, revised version, accepted for publication in J. Reine Angew. Math. (Crelle's

The main theorem of the author's thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin-Stark conjecture and Leopoldt's conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

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In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are 0 or 1 (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define (one unconditional and other conjectural) integral p-adic L-functions and cotorsion Selmer groups. This allows us to reformulate Perrin-Riou's main conjecture in terms of these objects, in the same fashion as Kobayashi's ±-Iwasawa theory for supersingular elliptic curves. By the aid of the theory of Coleman-adapted Kolyvagin systems we develop here, we deduce parts of Perrin-Riou's main conjecture from an explicit reciprocity conjecture.

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. Contents