Class Groups of Number Fields and Related Topics

✍ Scribed by Kalyan Chakraborty (editor), Azizul Hoque (editor), Prem Prakash Pandey (editor)

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 182
• Edition: 1
• Category: Library

✦ Synopsis

This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4–7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields and introduces new techniques and tools to study these problems. Topics in this book include class groups and class numbers of number fields, units, the Kummer–Vandiver conjecture, class number one problem, Diophantine equations, Thue equations, continued fractions, Euclidean number fields, heights, rational torsion points on elliptic curves, cyclotomic numbers, Jacobi sums, and Dedekind zeta values.

This book is a valuable resource for undergraduate and graduate students of mathematics as well as researchers interested in class groups of number fields and their connections to other branches of mathematics. New researchers to the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.

✦ Table of Contents

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