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Tuesday, November 10, 2020 | History

1 edition of Iwasawa theory, projective modules, and modular representations found in the catalog.

Iwasawa theory, projective modules, and modular representations

Ralph Greenberg

# Iwasawa theory, projective modules, and modular representations

Written in English

Subjects:
• Iwasawa theory,
• Elliptic Curves

• Edition Notes

Classifications The Physical Object Statement Ralph Greenberg Series Memoirs of the American Mathematical Society -- no. 992 LC Classifications QA247 .G74 2010 Pagination v, 185 p. : Number of Pages 185 Open Library OL25053531M ISBN 10 9780821849316 LC Control Number 2011002508 OCLC/WorldCa 701619879

In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles ().The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main tured by: Kenkichi Iwasawa. Student Iwasawa Theory Seminar Karl Schaefer and Eric Stubley the classi cation of (nitely-generated) -modules (up to quasi-isomorphism), and apply this to the module Bernoulli numbers, Eisenstein Series, Modular Forms, Galois Representations, Unrami ed Extensions, Class Groups. Time permitting, deduce the results of Herbrand and Ribet File Size: KB. This book gives an accessible introduction to Iwasawa theory. (Primary Text 2) Hida, H., "Hilbert modular forms and Iwasawa theory," Oxford mathematical monographs. Oxford: Clarendon, This book describes the use of modular forms and their associated Galois representations in Iwasawa theory. These notes include the main topics of a basic graduate algebra course: finite groups, the Sylow theorems, finite group actions, rings, modules, fields, Galois theory. In addition, the first set of notes include injective and projective limits of groups, while the second set of notes includes topics on homological algebra (derived functors.

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚ p, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields Cited by:

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### Iwasawa theory, projective modules, and modular representations by Ralph Greenberg Download PDF EPUB FB2

: Iwasawa Theory, Projective Modules, and Modular Representations (Memoirs of the American Mathematical Society: Volume ) (): Ralph Greenberg: Books.

This paper shows that properties of projective modules over a group ring $$\mathbf{Z}_p[\Delta]$$, where $$\Delta$$ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve $$E$$.

Modular representation theory for the group $$\Delta$$ plays a crucial role in this study. Iwasawa Theory, projective modules, and modular representations About this Title Ralph Greenberg, Department of Mathematics, University of Washington, Seattle, Washington There is a projective O[∆]-module Pτ which is characterized (up to isomorphism) as follows: Pτ has a unique maximal O[∆]-submodule and the corresponding quotient module is isomorphic to Uτ.

One often refers to Pτ as the projective hull of Uτ as an O[∆]-module. The Pτ’s are precisely the indecomposable, projective O[∆] Size: 1MB.

BibTeX @ARTICLE{Greenberg_iwasawatheory, author = {Ralph Greenberg}, title = {Iwasawa Theory, Projective Modules, and Modular Representations}, journal = {Memoirs of. The two main topics of this book are Iwasawa theory and modular forms.

The presentation of the theory of modular forms starts with several beautiful relations discovered by Ramanujan and leads to a discussion of several important ingredients, including the zeta-regularized products, And modular representations book limit formula, and the Selberg trace by: This paper shows that properties of projective modules over a group ring Z p[Δ], where Δ is a finite Galois group, can projective modules used to study the behavior of certain invariants which occur naturally in Projective modules Ralph Greenberg.

Techniques such as Hida’s theory of p-adic modular forms and big Galois representations play a crucial part. Also a noncommutative Iwasawa theory of arbitrary p-adic Lie extensions has been developed. This volume aims to present a snapshot of the state of art of Iwasawa theory as of Iwasawa Theory, projective modules, and modular representations - Ralph Greenberg, Department of Mathematics, University of Washington, Seattle, Washington Projective modules ρ1 is any representation of ∆ over F, then there exists a representation ρ2 of ∆ factoring through the quotient group ∆0 such that ρess 1 ∼= ρess 2.

Hence, under the assumption that X is a ﬁnitely-generated, projective Z p[∆]-module, one can determine λ X(σ) for all σ ∈ IrrF(∆) if. «Isotope Geology Iwasawa Theory, Projective Modules, and Modular Representations (Memoirs of the American Iwasawa theory Society)» Categories Adult Magazine ().

ELEMENTARY MODULAR IWASAWA THEORY 3 1. Curves over a field Any algebraic curve over an algebraically closed ﬁeld can be embedded into the 3-dimensional projective space P3 (e.g., [ALG, IV]) and any closed curve in P3 is birationally isomorphic to a curve inside P2 (a plane curve; see [ALG, IV]), we give some details of the theory of plain curve deﬁned over a ﬁeld k⊂C in this.

Introduction --Projective and quasi-projective modules --Projectivity or quasi-projectivity of X [sigma with 0 subscript, E] (K[infinity]) --Selmer atoms --The structure of Hv (K[infinity], E) --The case where [delta] is a p-group --Other specific groups --Some arithmetic illustrations --Self-dual representations --A duality theorem --p-modular functions --Parity --More arithmetic illustrations.

Abstract. This survey paper is focused on a connection between the geometry of $$\mathop{\text{GL}}\nolimits _{d}$$ and the arithmetic of $$\mathop{\text{GL}}\nolimits _{d-1}$$ over global fields, for integers d ≥ 2.

For d = 2 over $$\mathbb{Q}$$, there is an explicit and modular representations book of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is Cited by: 4. The two main topics of this book are Iwasawa theory and modular forms.

The presentation of the theory of modular forms starts with several beautiful relations discovered by Ramanujan and leads to a discussion of several important ingredients, including the zeta-regularized products, Kronecker's limit formula, and the Selberg trace formula. Iwasawa theory, projective modules, and modular representations (DLC) (OCoLC) Material Type: Document, Internet resource: Document Type: Internet Resource, Computer File: All Authors / Contributors: Ralph Greenberg.

arXivv1 [] 12 Nov Modular Representations Of Proﬁnite Groups JohnWilliamMacQuarrie October26, 1 Introduction The modular representation theory of a ﬁnite group Gattempts to describe the modules over the group algebra kG, where kis a ﬁeld of characteristic p dividing the order of G.

But the projective modules are more elusive (and in the general case remain so for small primes). Since the Steinberg module is simple as well as projective, one relies on the very general principle for finite dimensional Hopf algebras that tensoring any module with a projective one yields a projective module.

IwasawaTheory and Lubin-Tate (ϕ,Γ)-Modules Let us say that an F-linear representation V of GK is F-analytic if for all embeddings τ: F→Q p, with τ6= Id, the representation C ⊗τ FV is trivial (as a semilinear Cp-representation of GK).The following result is known [Ber16].

Theorem. If V is an F-analytic representation of GK, it is overconvergent. Another source of overconvergent. On μ-invariants of selmer groups of some cm elliptic curves Iwasawa Theory, Projective Modules and Modular Representations, Memoirs of the American Mathematical Society, Vol.

(American. In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa () (岩澤 健吉), as part of the theory of cyclotomic the early s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties.

Iwasawa Theory, Projective Modules, and Modular Representations, Ralph Greenberg,Memoirs of the American Mathematical Society, Memories of Professor Iwasawa, (Ralph Greenberg) minutes at the house of Professor Kenkichi Iwasawa (Ralph Greenberg).

modular-forms galois-representations iwasawa-theory. asked Nov 23 '19 at user 0. votes. 0answers An issue with showing that an Iwasawa module has zero $\mu$ invariant. Newest iwasawa-theory questions feed. Summary. The µ-invariant is a fundamental invariant in Iwasawa theory and a long-standing conjecture of Iwasawa asserts that it is zero for number can be viewed as a statement on an arithmetic Iwasawa module associated to the trivial motive.

In this chapter, we discuss what the analogous conjecture should be for elliptic by: 4. Iwasawa Theory for Coleman families as applications of this Coleman map.

Contents 1. Introduction 1 2. Iwasawa Main Conjecture for a cuspform (Review on classical results) 4 Selmer group and p-adic L-function 5 Iwasawa Main Conjecture 9 3.

Setting of Iwasawa theory for a family of cuspforms 16 Review of Hida families 17 This is the third of three related volumes on number theory. (The first two volumes were also published in the Iwanami Series in Modern Mathematics, as volumes and ) The two main topics of this book are Iwasawa theory and modular forms.

The presentation of the theory of modular forms starts with several beautiful relations discovered by Ramanujan and leads to a discussion of several. The study of the symmetric groups forms one of the basic building blocks of modern group theory. This book is the first completely detailed and self-contained presentation of the wealth of information now known on the projective representations of the symmetric and alternating groups.

Prerequisites are a basic familiarity with the elementary theory of linear representations and a modest. The work of Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt.

This book, authored by a leading researcher, describes the striking applications that. Romyar Sharifi - Modular symbols in Iwasawa theory. Abstract. I will explain a very explicit, conjectural relationship between first homology groups of modular curves modulo Eisenstein ideals and second K-groups of cyclotomic integer rings, in a form the mildly refines the published conjecture.

The aim of the present paper is to provide some evidence that, in accordance with the main conjectures of Iwasawa theory, there is a close connection between the action of the Selmer group of over, and the global root numbers attached to the twists of the complex -function of by Artin representations of.

Celebrating one of the leading figures in contemporary number theory – John H. Coates – on the occasion of his 70th birthday, this collection of contributions covers a range of topics in number theory, concentrating on the arithmetic of elliptic curves, modular forms, and Galois representations.

Root numbers and parity of local Iwasawa invariants. Journal of Number Theory, Vol. Issue., p. you will be asked to authorise Cambridge Core to connect with your account.

Find out more Iwasawa theory, projective modules, and modular representations, Preprint. [12] Harris, M., Shepherd-Barron, N.

and Taylor, R., A Cited by: In this paper, we will prove the non-commutative Iwasawa main conjecture—formulated by John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob ()—for certain specific non-commutative p-adic Lie extensions of totally real fields by using theory on integral logarithms introduced by Robert Oliver and Laurence R.

Taylor, theory on Hilbert modular forms introduced by Cited by: R. Greenberg, Algebraic Number Theory, Advanced Studies in Pure Mathematics 17 (Academic Press, Boston, MA, ) pp.

97– Crossref, Google Scholar; R. Greenberg, Iwasawa Theory, Projective Modules and Modular Representations, Memoirs of the American Mathematical Society (American Mathematical Society, Providence, RI, ).Author: Chandrakant S.

Aribam. Iwasawa theory for Artin representations I 3 Hypothesis A: The degree [K: Q] is not divisible by p. Also, d+ = 1. Furthermore, there exists a 1-dimensional representation εp of ∆p which occurs with multiplicity 1 in ρj∆ p.

Our theory depends on choosing one such character εp of ∆p. Also, if d>1, then Kmust be totally complex and hence. Iwasawa theory for modular forms at supersingular primes I will talk about joint work with Antonio Lei and David Loeﬄer.

We de-ﬁne a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Q p using the theory of Wach modules. Let. Reﬁned Iwasawa theory for p-adic representations and the structure of Selmer groups Masato Kurihara Dedicated to Peter Schneider on his 60th birthday Abstract In this paper, we develop the idea in [16] to obtain ﬁner results on the structure of Selmer modules for p-adic representations than the usual main conjecture in Iwasawa theory.

non abelian fundamental groups and iwasawa theory Download non abelian fundamental groups and iwasawa theory or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get non abelian fundamental groups and iwasawa theory book now.

This site is like a library, Use search box in the widget to get ebook. Background. In this paper, we shall study the arithmetic of a specific class of p-adic Galois representations: those associated to the symmetric squares of modular Galois representations are important for two reasons.

Firstly, they are very natural examples of global Galois representations, and thus provide a good testing ground for general conjectures relating arithmetic. —, Iwasawa theory, projective modules, and modular representations, in preparation. Harris, Systematic growth of Mordell-Weil groups of abelian varieties in towers of number fields, Invent.

Math. 51 (), Cited by:. Igusa curves Fix a prime p. For r ≥0, the Igusa curve Ig(pr) of level pr is the moduli space of pairs (E,Q) E = A generalized elliptic curve Q = A point of E(pn)generating the kernel of Vn: E(pn →E Ig(pr) is a smooth projective curve /Fp, of genus ∼prϕ(pr) There are natural quotient maps.Modular symbols in Iwasawa theory Takako Fukaya, Kazuya Kato, and Romyar Shariﬁ Novem 1 Introduction The starting point of this paper is the fascinatingly simple and explicit map [u: v]7!f1 u N,1 v N g that relates the worlds of geometry/topology and arithmetic [Bu, Sh].

Here, for N 1,Cited by: 4. AbstractThis is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic ℤ p {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits by: 3.