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

Abstract

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

Integral Iwasawa theory of galois representations for non-ordinary primes

“…Proof. The proof of this proposition is similar to the proof of Proposition 9.2 in [BL15]. Let F can denote the canonical Selmer structure on T given with the data H 1 Fcan (F λ , T) = H 1 (F λ , T) for every prime λ ∈ Σ .…”

On the Iwasawa theory of CM fields for supersingular primes

“…This work in part relies on the techniques developed here, as well as a general theory of plus/minus Coleman maps we develop in [BL16]. Although in [BL15], the authors are able to lift the hypotheses on Theorem B and C that p splits completely in F + /Q, they are able to deduce only one of the signed main conjectures (whereas we prove both main conjectures simultaneously here). Note that we could have also formulated 2 g signed main conjectures (as opposed to a single plus/minus main conjecture) here as well, by assigning one of the "plus" or "minus local conditions" at each prime lying above p (as opposed assigning the "plus" or "minus local condition" everywhere above p uniformly) and prove each of them.…”

“…We hope that this will allow us to verify the explicit reciprocity conjectures (and therefore deduce our main results here unconditionally) in the situation of Remark 1.2, namely when F + (E[p])/K is abelian. For the time being, this does not seem tractable in the rather abstract set up of [BL15].…”

On the parity of supersingular Weil polynomials

“…We recall that a multi-signed main conjecture for abelian varieties with supersingular reduction has been formulated in [BL15b], generalizing the results in [Kob03,Spr12]. Furthermore, this conjecture has been proved in [BL15a] under the hypothesis that certain Rubin-Stark elements from [Rub92,Rub96] exist. The main conjecture in [BL15b] relies on the existence of a logarithmic matrix that is used to decompose Perrin-Riou's (conjectural) p-adic L-function from [PR95] and to define the appropriate signed Selmer groups.…”