Number Theory 3: Iwasawa Theory and Modular Forms

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
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