Stark units and the main conjectures for totally real fields

Proc. London Math. Soc.

Lei

2015

Self Cite

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

Section: Rubin-stark Element Euler System Of Rank Gmentioning

confidence: 99%

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

Proc. London Math. Soc.

Lei

2015

Self Cite

“…with the property that if c maps to κ κ κ ∈ KS(T, F Λ , P) then Remark 3.8. It may be proved that the Λ-adic Kolyvagin system κ κ κ χ which here we have constructed out of the (conjectural) Rubin-Stark elements does exist unconditionally, using the techniques of [Büy11,Büy12]; see also [Büy09b,Theorem 2.19].…”

Section: Choosing the Homomorphismsmentioning

confidence: 99%

Main Conjectures for CM Fields and a Yager-Type Theorem for Rubin–Stark Elements

International Mathematics Research Notices

2013

Self Cite

In this article, we study the p-ordinary Iwasawa theory of the (conjectural) Rubin-Stark elements defined over abelian extensions of a CM field F and develop a rank-g Euler/Kolyvagin system machinery (where 2g = [F : Q]), refining and generalizing Perrin-Riou's theory and the author's prior work. This has several important arithmetic consequences: Using the recent results of Hida and Hsieh on the CM main conjectures, we prove a natural extension of a theorem of Yager for the CM field F , where we relate the Rubin-Stark elements to the several-variable Katz p-adic L-function. Furthermore, beyond the cases covered by Hida and Hsieh, we are able to reduce the p-ordinary CM main conjectures to a local statement about the Rubin-Stark elements. We discuss applications of our results in the arithmetic of CM abelian varieties.

“…Besides the standard applications (i.e., Theorem A and Theorem B above) of our construction of what we call an L L L-restricted Kolyvagin system (c.f., Definition 2.36, Theorem 4.15 and Remark 4.8) out of Stickelberger elements, we also prove the following statement regarding the local Iwasawa theory of Rubin-Stark elements, which was proved in [Büy09b] assuming the truth of the main conjecture:…”

mentioning

confidence: 68%

“…The G k -representation T turns out to have core Selmer rank r as well. As in [Büy09a,Büy09b], we introduce certain modified Selmer groups associated with the representation T . In this setting, the Euler system that gives rise to the Kolyvagin system which controls the modified Selmer group is obtained from the Stickelberger elements.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Stickelberger elements and Kolyvagin systems

2011

Nagoya Math. J.

In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are two-fold: First, assuming Brumer's conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce Iwasawa's main conjecture for totally real fields (for totally odd characters). Although this portion of our results have already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles' work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler / Kolyvagin system machinery when the core Selmer rank is r > 1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the 'rigidity' of the collection of Kolyvagin systems proved by Mazur, Rubin and the author. Assumption 1.1. θ χ K annihilates A χ K . We remark here that Wiles [Wil90a] proved that Brumer's conjecture as stated above follows from his proof [Wil90b] of the main conjecture of Iwasawa theory for totally real fields. In this paper, we prove the other way around, namely that, assuming Brumer's conjecture, one might also prove the main conjecture (see Theorem B below; see also Kurihara's work [Kur03] where he refines Wiles' result using a different type of Euler system argument).The first application of the treatment here is the following (Theorem 5.3 below):Theorem A. Suppose the hypothesis (A1) and Assumption 1.1 hold. Then,With a bit more work, we can prove the Iwasawa theoretic version of Theorem A, which we state below. Set Γ = Gal(k ∞ /k) and Λ = Z p [[Γ]], as usual. Let L ωχ −1 ∈ Λ denote the Deligne-Ribet p-adic L-function (see [DR80]). We recall in (5.2) the basic interpolation property which characterizes L ωχ −1 . Let Tw ρcyc be a certain twisting operator on Λ (see §5.2 below for its definition). For any abelian group A, let A ∨ := Hom(A, Q p /Z p ) denote its Pontryagin dual, and finally, let char(M) denote the characteristic ideal of a finitely generated Λ-module M (with the convention that char(M) = 0 unless M is Λ-torsion). Then we are able to prove (see Theorem 5.8 and Corollary 5.10 for a slightly improved version so as to include the case µ µ µ p ⊂ L):Theorem B. Suppose the hypotheses (A1)-(A2) as well as Assumption 1.1 hold. Assume also that L does not contain pth roots of unity. Then, Proposition 2.25. We have X (T, F can ) = r (= [k : Q]).Proof. This follows from [MR04, Theorem 5.2.15], applied with the base field k (instead of Q; we therefore have r real places instead of one) and using our assumption that χ is totally odd.Proposition 2.26. The core Selmer rank X (T, F L ) of the Selmer structure F L on T is one.