Class Groups of Number Fields and Related Topics

Preface
Contents
About the Editors
A Geometric Approach to Large Class Groups: A Survey
	1 The Survey
		1.1 Large Class Groups: The Folklore Conjecture
		1.2 A Toy Example
		1.3 General Specialization Results
		1.4 Record of Known Results Towards Conjecture 1.1
	2 The Examples
		2.1 Yamamoto\'s Result
		2.2 3-Ranks of Quadratic Fields: A Construction of Craig
		2.3 5-Ranks of Quadratic Fields: A Construction of Mestre
		2.4 Higher Degree Fields
	References
On Simultaneous Divisibility of the Class Numbers of Imaginary Quadratic Fields
	1 Introduction
	2 Old Motivation for the Results
	3 Comparison of Methods
	4 Construction of Fields and Extensions
	5 Construction of Fields and Ideals
	6 New Motivation, Application to a Problem
	7 Real Quadratic Cases
	References
Thue Diophantine Equations
	1 Thue Equations
		1.1 Introduction
		1.2 Positive Definite Binary Forms
		1.3 Thue Equation and Diophantine Approximation
		1.4 An Example: x3-2y3=m
	2 Solving Thue Equation Using Baker\'s Method
		2.1 References
		2.2 Thue Equation and Siegel\'s Unit Equation
		2.3 Lower Bounds for Linear Forms in Logarithms  and Siegel\'s Unit Equation
	3 Families of Thue Equations
		3.1 Historical Survey
		3.2 Idea of the Proof
		3.3 Joint Papers with Claude Levesque
	4 A Guide to Further References
	References
A Lower Bound for the Class Number  of Certain Real Quadratic Fields
	1 Introduction
	2 A Lower Bound for the Class Number
	3 Proof of Theorem 1
	4 A Sequence langle2,…, 2,2, 1rangle of Pre-ELE2 Type
	References
A Survey of Certain Euclidean Number Fields
	1 Introduction
	2 Explicit Construction of Potentially Euclidean Real Quadratic Fiel
	3 The Cubic Case
	References
Divisibility of Class Number of a Real Cubic or Quadratic Field and Its Fundamental Unit
	1 Introduction
	2 Fundamental Unit of mathbbQ(sqrt[3]m) when 3hm
	3 Real Quadratic Fields with Odd Class Number
	References
The Charm of Units I,  On the Kummer–Vandiver Conjecture. Extended Abstract
	1 Introduction
		1.1 Plan of the Proof
		1.2 Notations and Auxiliary Results
		1.3 Hilbert\'s Theorems on Class Fields
	2 Primes and Local Units
		2.1 Primes Above p
		2.2 Local Units in Fields and p-idèles
	3 Existence of a Singular Capitulation Unit
	4 Proof of the Main Theorem
	References
Heights and Principal Ideals of Certain Cyclotomic Fields
	1 Introduction
	2 Heights
	3 Discriminant Bounds
	4 Plans\' Theorem
	References
Distribution of Residues Modulo p Using the Dirichlet\'s Class Number Formula
	1 Introduction
	2 Preliminaries
	3 Proof of Theorem 1
	4 Proof of Theorem 2
	5 Proof of Theorem 3
	References
On Class Number Divisibility of Number Fields and Points on Elliptic Curves
	1 Introduction
	2 Class Number Related Questions
	3 Homomorphisms from the Group of Rational Points on Elliptic Curves to Class Group of Number Fields
	4 A Construction for Biquadratic Fields of Even Class Number
	References
Small Fields with Large Class Groups
	1 Introduction
	2 Proof of the Theorem
	References
Cyclotomic Numbers and Jacobi Sums: A Survey
	1 Introduction
	2 Definitions and Notations
	3 Properties of Jacobi Sums and Cyclotomic Numbers
	4 Jacobi Sums and It Congruences
	5 Cyclotomic Numbers
	6 Concluding Remarks
	References
A Pair of Quadratic Fields with Class Number Divisible by 3
	1 Introduction
	2 Some Useful Results
	3 Proof of the Theorem 1.1
	References
On Lebesgue–Ramanujan–Nagell Type Equations
	1 Introduction
	2 The Equation x2+Dm=yn
	3 The Equation x2+Dm=2yn
	4 On the Equation x2+Dm=4yn
	References
Partial Dedekind Zeta Values and Class Numbers of R–D Type Real Quadratic Fields
	1 Introduction
	2 R–D Type Real Quadratic Fields and Some Conjectures
	3 Dedekind Zeta Values
	4 Class Number Criteria
	References
On the Continued Fraction Expansions  of sqrtp and sqrt2p for Primes pequiv38mu(mod6mu4)
	1 Introduction
	2 Continued Fraction Expansions of Quadratic Irrationalities
	3 Proof of Theorem1
	References