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Euler Systems. (AM-147), Volume 147 : Annals of Mathematics - Karl Rubin

Euler Systems. (AM-147), Volume 147

Annals of Mathematics


Published: 1st May 2000
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One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980s in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field.

Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations.

The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Acknowledgmentsp. xi
Introductionp. 3
Notationp. 6
Galois Cohomology of p-adic Representationsp. 9
p-adic Representationsp. 9
Galois Cohomologyp. 11
Local Cohomology Groupsp. 12
Local Dualityp. 18
Global Cohomology Groupsp. 21
Examples of Selmer Groupsp. 23
Global Dualityp. 28
Euler Systems: Definition and Main Resultsp. 33
Euler Systemsp. 33
Results over Kp. 36
Results over K,,p. 40
Twisting by Characters of Finite Orderp. 43
Examples and Applicationsp. 47
Preliminariesp. 47
Cyclotomic Unitsp. 48
Elliptic Unitsp. 55
Stickelberger Elementsp. 55
Elliptic Curvesp. 63
Symmetric Square of an Elliptic Curvep. 73
Derived Cohomology Classesp. 75
Setupp. 75
The Universal Euler Systemp. 78
Properties of the Universal Euler Systemp. 80
Kolyvagin's Derivative Constructionp. 83
Local Properties of the Derivative Classesp. 90
Local Behavior at Primes Not Dividing prp. 92
Local Behavior at Primes Dividing rp. 98
The Congruencep. 102
Bounding the Selmer Groupp. 105
Preliminariesp. 105
Bounding the Order of the Selmer Groupp. 106
Bounding the Exponent of the Selmer Groupp. 114
Twistingp. 119
Twisting Representationsp. 119
Twisting Cohomology Groupsp. 121
Twisting Euler Systemsp. 122
Twisting Theoremsp. 125
Examples and Applicationsp. 125
Iwasawa, Theoryp. 129
Overviewp. 129
Galois Groups and the Evaluation Mapp. 135
Proof of Theorem 2.3.2p. 141
The Kernel and Cokernel of the Restriction Mapp. 145
Galois Equivariance of the Evaluation Mapsp. 147
Proof of Proposition 7.1.7p. 151
Proof of Proposition 7.1.9p. 154
Euler Systems and p-adic L-functionsp. 163
The Settingp. 164
Perrin-Riou's p-adic L-function and Related Conjecturesp. 166
Connection with Euler Systems when d- = 1p. 168
Example: Cyclotomic Unitsp. 171
Connection with Euler Systems when d- >1p. 173
Variantsp. 175
Rigidityp. 175
Finite Primes Splitting Completely in K,,,IKp. 178
Euler Systems of Finite Depthp. 179
Anticyclotomic Euler Systemsp. 180
Additional Local Conditionsp. 183
Varying the Euler Factorsp. 185
Linear Algebrap. 189
Herbrand Quotientsp. 189
p-adic Representationsp. 191
Continuous Cohomology and Inverse Limitsp. 195
Preliminariesp. 195
Continuous Cohomologyp. 195
Inverse Limitsp. 198
Induced Modulesp. 201
Semilocal Galois Cohomologyp. 202
Cohomology of p-adic Analytic Groupsp. 205
Irreducible Actions of Compact Groupsp. 205
Application to Galois Representationsp. 207
p-adic Calculations in Cyclotomic Fieldsp. 211
Local Units in Cyclotomic Fieldsp. 211
Cyclotomic Unitsp. 216
Bibliographyp. 219
Index of Symbolsp. 223
Subject Indexp. 227
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691050768
ISBN-10: 0691050767
Series: Annals of Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 240
Published: 1st May 2000
Country of Publication: US
Dimensions (cm): 23.5 x 15.88  x 1.91
Weight (kg): 0.34