Iwasawa Theory and Its Perspective, Volume 2

Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to -adic -functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory.The goal of this second part of the three-part publication is to explain various aspects of the cyclotomic Iwasawa theory of -adic Galois representations.ReadershipGraduate students and researchers interested in number theory and arithmetic geometry. Cover Title page Contents Preface to the second volume Preface to the first volume Chapter 4. Introduction to cyclotomic Iwasawa theory of elliptic curves 4.1. Elliptic curves and Selmer groups 4.2. Hasse–Weil L-function and p-adic L-function of an elliptic curve 4.3. BSD conjecture and the finiteness of Selmer groups 4.4. Cyclotomic Iwasawa main conjecture of an ordinary elliptic curve Chapter 5. Framework of cyclotomic Iwasawa theory for p-adic Galois representations 5.1. Setting 5.2. Definition of various types of Selmer groups 5.3. Fundamental Conjectures A, B, and C Chapter 6. Known results on cyclotomic Iwasawa theory for p-adic representations 6.1. Known results in the rank-1 case 6.2. Known results in the rank-2 case on the analytic side 6.3. Known results in the rank-2 case for the Iwasawa main conjecture 6.4. Construction of the Beilinson–Kato Euler system 6.5. Iwasawa main conjecture via the method of Euler systems 6.6. Generalization to higher-rank cases and related topics Appendix A. A.1. Review of modular curves and modular forms A.2. Review of motives A.3. Review of p-adic Hodge theory References Index Back Cover