Euler Systems. (AM-147).
Title:
Euler Systems. (AM-147).
Author:
Rubin, Karl.
ISBN:
9781400865208
Personal Author:
Physical Description:
1 online resource (241 pages)
Series:
Annals of Mathematics Studies ; v.147
Annals of Mathematics Studies
Contents:
Cover -- Title -- Copyright -- Contents -- Acknowledgments -- Introduction -- Notation -- Chapter 1. Galois Cohomology of p-adic Representations -- 1.1. p-adic Representations -- 1.2. Galois Cohomology -- 1.3. Local Cohomology Groups -- 1.4. Local Duality -- 1.5. Global Cohomology Groups -- 1.6. Examples of Selmer Groups -- 1.7. Global Duality -- Chapter 2. Euler Systems: Definition and Main Results -- 2.1. Euler Systems -- 2.2. Results over K -- 2.3. Results over K∞ -- 2.4. Twisting by Characters of Finite Order -- Chapter 3. Examples and Applications -- 3.1. Preliminaries -- 3.2. Cyclotomic Units -- 3.3. Elliptic Units -- 3.4. Stickelberger Elements -- 3.5. Elliptic Curves -- 3.6. Symmetric Square of an Elliptic Curve -- Chapter 4. Derived Cohomology Classes -- 4.1. Setup -- 4.2. The Universal Euler System -- 4.3. Properties of the Universal Euler System -- 4.4. Kolyvagin's Derivative Construction -- 4.5. Local Properties of the Derivative Classes -- 4.6. Local Behavior at Primes Not Dividing pτ -- 4.7. Local Behavior at Primes Dividing τ -- 4.8. The Congruence -- Chapter 5. Bounding the Selmer Group -- 5.1. Preliminaries -- 5.2. Bounding the Order of the Selmer Group -- 5.3. Bounding the Exponent of the Selmer Group -- Chapter 6. Twisting -- 6.1. Twisting Representations -- 6.2. Twisting Cohomology Groups -- 6.3. Twisting Euler Systems -- 6.4. Twisting Theorems -- 6.5. Examples and Applications -- Chapter 7. Iwasawa Theory -- 7.1. Overview -- 7.2. Galois Groups and the Evaluation Map -- 7.3. Proof of Theorem 2.3.2 -- 7.4. The Kernel and Cokernel of the Restriction Map -- 7.5. Galois Equivariance of the Evaluation Maps -- 7.6. Proof of Proposition 7.1.7 -- 7.7. Proof of Proposition 7.1.9 -- Chapter 8. Euler Systems and p-adic L-functions -- 8.1. The Setting -- 8.2. Perrin-Riou's p-adic L-function and Related Conjectures.
8.3. Connection with Euler Systems when d_ = 1 -- 8.4. Example: Cyclotomic Units -- 8.5. Connection with Euler Systems when d_ > 1 -- Chapter 9. Variants -- 9.1. Rigidity -- 9.2. Finite Primes Splitting Completely in K∞/ K -- 9.3. Euler Systems of Finite Depth -- 9.4. Anticyclotomic Euler Systems -- 9.5. Additional Local Conditions -- 9.6. Varying the Euler Factors -- Appendix A. Linear Algebra -- A.1. Herbrand Quotients -- A.2. p-adic Representations -- Appendix B. Continuous Cohomology and Inverse Limits -- B.1. Preliminaries -- B.2. Continuous Cohomology -- B.3. Inverse Limits -- B.4. Induced Modules -- B.5. Semilocal Galois Cohomology -- Appendix C. Cohomology of p-adic Analytic Groups -- C.1. Irreducible Actions of Compact Groups -- C.2. Application to Galois Representations -- Appendix D. p-adic Calculations in Cyclotomic Fields -- D.1. Local Units in Cyclotomic Fields -- D.2. Cyclotomic Units -- Bibliography -- Index of Symbols -- Subject Index.
Abstract:
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.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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