Kazim Buyukboduk

Kolyvagin systems of Stark units

Abstract In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier 46: 33–62, 1996], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mazur and Rubin, Mem. Amer. Math. Soc. 168: viii, 96, 2004]) 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: 141–154, 1992] (in a sense we explain below), and of [Perrin-Riou, Ann. Inst. Fourier 48: 1231–1307, 1998], [Rubin, Euler systems, Princeton University Press, 2000] applied to our setting.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].

Λ-adic Kolyvagin Systems

In this paper we study Kolyvagin Systems, as defined in [MR04], over the cyclotomic Zp-tower for a Gal(Q/Q) representation T , which is free of finite rank over Zp. We prove, under certain hypotheses, that the module Kolyvagin Systems over Λ is free of rank one over the (cyclotomic) Iwasawa algebra Λ. We include a conjectural picture, in the spirit of [PR95] and [Rub00] §8, about how to relate the Kolyvagin Systems to p-adic L-functions. We also study the Iwasawa theory of Stark units via the perspective offered by our main theorem, and provide a strategy to deduce the main conjectures of Iwasawa theory for totally real number fields.

Deformations of Kolyvagin systems

Mazur and Rubin prove the existence of Kolyvagin systems fo r a general class of modp Galois representations̄ ρ. Furthermore, they also prove (under certain hypotheses) t hat these Kolyvagin systems may be deformed to Kolyvagin system for a deformation of̄ ρ to a discrete valuation ring. The goal of this article is to achie ve this for larger coefficient rings. More specifically, we carry this out for two types of deformat ions of Galois representations: the universal deformation when the deformation problem is u nobstructed, and deformations to a two-dimensional Gorenstein ring. This generalizes the wo rks f Howard on Heegner points and Ochiai on Kato’s Euler system. We give explicit arithmet ic applications of our result on the existence of ‘big’ Kolyvagin systems, such as the interp olation of Kato’s Euler system in families (not necessarily p-ordinary) of modular Galois representations.

Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.

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

In this article, we study the p-ordinary Iwasawa theory of the (conjectural) RubinStark elements defined over abelian extensions of a CM field F and develop a rankg Euler/Kolyvagin system machinery (where 2g = [F : Q]), refining and generalizing Perrin-Riou’s theory and the author’s prior work. This has several importa nt arithmetic consequences: Using the recent results of Hida and Hsieh on the CM main conjecture s, w 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 Katzp-adicL-function. Furthermore, beyond the cases covered by Hida an 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 resul ts in the arithmetic of CM abelian varieties.

Stark units and the main conjectures for totally real fields

Main theorem of [Büy07b] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Büy07a] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank 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 of local Iwasawa theory. This statement, however, turns out to be interesting in its own right as it suggests a relation between solutions to p-adic and complex Stark conjectures.

Integral Iwasawa theory of galois representations for non-ordinary primes

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 integral p-adic L-functions and (one unconditionally and other conjecturally) 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.

On Euler systems of rank

In this paper we set up a general Kolyvagin system machinery for Euler systems of rank r (in the sense of Perrin-Riou) associated to a large class of G alois representations, building on our previous work on Kolyvagin systems of RubinStark units and generalizing the results of Kato, Rubin and Perrin-Riou. Our machinery produ ces a bound on the size of the classical Selmer group attached to a Galoýs representatio T (that satisfies certain technical hypotheses) in terms of a certain r × r determinant; a bound which remarkably goes hand in hand with Bloch-Kato conjectures. At the end, we present an a pplication based on a conjecture of Perrin-Riou onp-adicL-functions, which lends further evidence to Bloch-Kato con jectures. CONTENTS

r

The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field

and their Kolyvagin systems

F