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Iwasawa algebra

About: Iwasawa algebra is a research topic. Over the lifetime, 174 publications have been published within this topic receiving 4044 citations.

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01 Jan 2000
TL;DR: Part I algebraic theory: Cohomology of Profinite groups as mentioned in this paper, some homological algebra, duality properties of profinite groups, free products of finite groups, Iwasawa Modules.
Abstract: Part I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules.- Part II Arithmetic Theory: Galois Cohomology.- Cohomology of Local Fields.- Cohomology of Global Fields.- The Absolute Galois Group of a Global Field.- Restricted Ramification.- Iwasawa Theory of Number Fields.- Anabelian Geometry.- Literature.- Index.

948 citations

01 Dec 1976
TL;DR: In this paper, the Eichler-Shimura Isomorphism on SL2(Z) has been used to define a modified version of the Petersson Scalar Product.
Abstract: I. Classical Theory.- I. Modular Forms.- 1. The Modular Group.- 2. Modular Forms.- 3. The Modular Function j.- 4. Estimates for Cusp Forms.- 5. The Mellin Transform.- II. Hecke Operators.- 1. Definitions and Basic Relations.- 2. Euler Products.- III. Petersson Scalar Product.- 1. The Riemann Surface ?\?.- 2. Congruence Subgroups.- 3. Differential Forms and Modular Forms.- 4. The Petersson Scalar Product.- Appendix by D. Zagier. The Eichler-Selberg Trace Formula on SL2(Z).- II. Periods of Cusp Forms.- IV. Modular Symbols.- 1. Basic Properties.- 2. The Manin-Drinfeld Theorem.- 3. Hecke Operators and Distributions.- V. Coefficients and Periods of Cusp Forms on SL2(Z).- 1. The Periods and Their Integral Relations.- 2. The Manin Relations.- 3. Action of the Hecke Operators on the Periods.- 4. The Homogeneity Theorem.- VI. The Eichler-Shimura Isomorphism on SL2(Z).- 1. The Polynomial Representation.- 2. The Shimura Product on Differential Forms.- 3. The Image of the Period Mapping.- 4. Computation of Dimensions.- 5. The Map into Cohomology.- III. Modular Forms for Congruence Subgroups.- VII. Higher Levels.- 1. The Modular Set and Modular Forms.- 2. Hecke Operators.- 3. Hecke Operators on q-Expansions.- 4. The Matrix Operation.- 5. Petersson Product.- 6. The Involution.- VIII. Atkin-Lehner Theory.- 1. Changing Levels.- 2. Characterization of Primitive Forms.- 3. The Structure Theorem.- 4. Proof of the Main Theorem.- IX. The Dedekind Formalism.- 1. The Transformation Formalism.- 2. Evaluation of the Dedekind Symbol.- IV. Congruence Properties and Galois Representations.- X. Congruences and Reduction mod p.- 1. Kummer Congruences.- 2. Von Staudt Congruences.- 3. q-Expansions.- 4. Modular Forms over Z[1/2, 1/3].- 5. Derivatives of Modular Forms.- 6. Reduction mod p.- 7. Modular Forms mod p, p?5.- 8. The Operation of ? on M?.- XI. Galois Representations.- 1. Simplicity.- 2. Subgroups of GL2.- 3. Applications to Congruences of the Trace of Frobenius.- Appendix by Walter Feit. Exceptional Subgroups of GL2.- V. p-Adic Distributions.- XII. General Distributions.- 1. Definitions.- 2. Averaging Operators.- 3. The Iwasawa Algebra.- 4. Weierstrass Preparation Theorem.- 5. Modules over Zp[[T]].- XIII. Bernoulli Numbers and Polynomials.- 1. Bernoulli Numbers and Polynomials.- 2. The Integral Distribution.- 3. L-Functions and Bernoulli Numbers.- XIV. The Complex L-Functions.- 1. The Hurwitz Zeta Function.- 2. Functional Equation.- XV. The Hecke-Eisenstein and Klein Forms.- 1. Forms of Weight 1.- 2. The Klein Forms.- 3. Forms of Weight 2.

477 citations

Journal ArticleDOI
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1986, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. implique l’accord avec les conditions générales d’utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

268 citations

Journal ArticleDOI
TL;DR: In this paper, the authors prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of a compact p-adic Lie group, with no element of order p, having a closed normal subgroup H such that G/H is isomorphic to Zp.
Abstract: Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z p. We prove the existence of a canonical Ore set S* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.

230 citations

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