-adic Kolyvagin Systems

International Mathematics Research Notices

2013

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.

Section: Introductionmentioning

confidence: 72%

“…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

“…In fact, we are able to prove considerably more than Theorem C in regard of the Rubin-Stark elements. In Theorem 5.16(i) below, we obtain a relation between the Stickelberger elements and Rubin-Stark elements (note that the existence of the latter is conjectural), making use of the formalism of L L L-restricted Euler systems we develop in this paper; as well as the rigidity of the module of Λ-adic Kolyvagin systems proved in [Büy10a]. This, we believe, is interesting on its own right.…”

mentioning

confidence: 77%

“…The Galois representation (and the Euler system attached to it) which we treat in this paper needs to be handled in a slightly different manner than the case of Rubin-Stark elements (which was studied in [Büy09a,Büy09b]), as far as the Euler / Kolyvagin system machinery is concerned. In a forthcoming paper [Büy10b], the set up from the current article and that from [Büy10a] are combined together to treat the theory of Kolyvagin systems which descend from Euler systems 1 for an arbitrary self-dual Galois representation whose core Selmer rank is r > 1.…”

mentioning

confidence: 99%

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.

“…This is one of the fundamental additions to the tools utilized in prior works [MR04,Büy11] on the general theory of Kolyvagin systems.…”

Section: Universal Kolyvagin Systems and Beilinson-kato Elementsmentioning

confidence: 98%

Deformations of Kolyvagin systems

2016

Ann. Math. Québec

ABSTRACT. Ochiai has previously proved that the Beilinson-Kato Euler systems for modular forms interpolate in nearly-ordinary p-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai's work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson-Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus) and use what we call universal Kolyvagin systems to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a p-adic L-function (in a certain sense of the word, in 3-variables).