Number Theory 3: Iwasawa Theory and Modular Forms (Translations of Mathematical Monographs) (Translations of Mathematical Monographs, 242)

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 186 and 240.) 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. The presentation of Iwasawa theory focuses on the Iwasawa main conjecture, which establishes far-reaching relations between a $p$-adic analytic zeta function and a determinant defined from a Galois action on some ideal class groups. This book also contains a short exposition on the arithmetic of elliptic curves and the proof of Fermat's last theorem by Wiles. Together with the first two volumes, this book is a good resource for anyone learning or teaching modern algebraic number theory. Cover Title page Contents Contents, Number Theory 2 Contents, Number Theory 1 Preface Preface to the English Edition Objectives and Outline of These Books Chapter 9. Modular Forms 9.1. Ramanujan’s discoveries (a) Mordell’s proof (b) Ramanujan’s congruence relation (c) Ramanujan’s identities and Lambert series (d) Ramanujan’s notebooks (e) What comes after the Ramanujan conjecture 9.2. Ramanujan’s Δ and holomorphic Eisenstein series (a) Holomorphic Eisenstein series (b) Relation between Δ and holomorphic Eisenstein series 9.3. Automorphy and functional equations (a) Wilton’s result (b) Hecke’s converse theorem 9.4. Real analytic Eisenstein series (a) Fundamental properties of E(s, z) (b) Application of the real analytic Eisenstien series (From GL(2) to GL(1)) Remark on the methodology (c) Rankin-Selberg method 9.5. Kronecker’s limit formula and regularized products (a) Regularized product (b) Determinant expression (c) Transformation formula for Δ (d) Transformation formula of E_2 (e) Calculation of Δ(i) and E_4(i) 9.6. Modular forms for SL_2(Z) (a) Fundamental properties of SL_2(Z) (b) Holomorphic modular forms (c) Real analytic modular forms 9.7. Classical modular forms (a) The case of congruence subgroups (b) Siegel modular forms Summary Exercises Chapter 10. Iwasawa Theory 10.0. What is Iwasawa theory (a) Ideal class group of cyclotomic fields (b) Herbrand and Ribet’s theorem (c) The ideal class group of K(μ_{p^n}) (d) Analogy between number fields and function fields (e) Action of Galois group 10.1. Analytic p-adic zeta function (a) Special values of the Riemann ζ function — gateway to the p-adic world (b) p-adic L-functions (c) Iwasawa function (d) Group algebra and completed group algebra (e) Galois group and p-adic L-function (f) Proof of Theorem 10.14 — Euler’s method and p-adic analogue (g) Proof of the Ferrero-Washington theorem 10.2. Ideal class groups and cyclotomic Zp-extensions (a) Power series and λ, μ invariants (b) Characteristic ideal and determinant (c) Proof of Proposition 10.23 (d) Maximal unramified abelian extension and ideal class group (e) Ideal class group of cyclotomic Z_p-extension (f) Proof of Theorem 10.25 and its applications (g) The minus part of the ideal class groups of abelian fields 10.3. Iwasawa main conjecture (a) Formulation of the Iwasawa main conjecture (b) Ideal class group of Q(μ_{p^∞}) (c) The χ part of the ideal class groups and the special values of ζ functions (d) Stickelberger’s theorem (e) Relation to modular forms (f) Iwasawa main conjecture for the plus part Summary Exercises Chapter 11. Modular forms (II) 11.1. Automorphic forms and representation theory (a) Three expressions of τ (n) and representation theory (b) From automorphic forms to automorphic representations 11.2. Poisson summation formula (a) The origin of the Poisson summation formula (b) Generalized Poisson summation formula (c) An application of the Poisson summation formula: ζ integral 11.3. Selberg trace formula (a) From the Poisson summation formula to the Selberg trace formula (b) The first application of the Selberg trace formula: the trace formula for Hecke operators (c) The second application of the Selberg trace formula: the Selberg ζ function 11.4. Langlands conjectures Summary Chapter 12. Elliptic curves (II) 12.1. Elliptic curves over the rational number field (a) Rational points over finite fields (b) Reduction mod l (c) n-torsion points and an action of Galois group (d) Tate module (e) ζ function and L-function of an elliptic curve (f) Modular elliptic curves 12.2. Fermat’s Last Theorem (a) Frey curve (b) Ribet’s theorem (c) Lift of modular Galois representations (d) R = T Summary Bibliography Number fields and algebraic number theory Elliptic curves ζ functions Class field theory Modular forms Iwasawa theory Answers to Questions Chapter 10 Chapter 12 Answers to Exercises Chapter 9 Chapter 10 Index Back Cover Nobushige Kurakawa, Masato Kurihara, Takeshi Saito ; Translated From The Japanese By Masato Kuwata. Originally Published In Japanese As V. 3 Of Sūron By K. Kato. Includes Bibliographical References (p. 211-215) And Index.