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 47, 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 KummerVandiver 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 researchersto the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.

Presents geometrical and classical techniques to study the class groups of number fields Features a chapter on KummerVandiver conjecture Explores various techniques to work toward Gauss class number one problem Includes a survey on LebesgueRamanujanNagell type equations Provides a rich history of cyclotomic numbers and Jacobi sums

Autorentext
KALYAN CHAKRABORTY is Professor at Harish-Chandra Research Institute (HRI), Allahabad, India, where he also obtained his Ph.D. in Mathematics. Professor Chakraborty was a postdoctoral fellow at IMSc, Chennai, and at Queen's University, Canada, and a visiting scholar at the University of Paris VI, VII, France; Tokyo Metropolitan University, Japan; Universitá Roma Tre, Italy; The University of Hong Kong, Hong Kong; Northwest University and Shandong University, China; Mahidol University, Thailand; Mandalay University, Myanmar; and many more. His broad area of research is number theory, particularly class groups, Diophantine equations, automorphic forms, arithmetic functions, elliptic curves, and special functions. He has published more than 60 research articles in respected journals and two books on number theory, and has been on the editorial boards of various leading journals. Professor Chakraborty is Vice-President of the Society for Special Functions and their Applications.
AZIZUL HOQUE is a national postdoctoral fellow at Harish-Chandra Research Institute (HRI), Allahabad. He earned his Ph.D. in Pure Mathematics from Gauhati University, Guwahati, in 2015. Before joining HRI, Dr. Hoque was Assistant Professor at the Regional Institute of Science and Technology, Meghalaya, and at the University of Science and Technology, Meghalaya. He has visited Hong Kong University, Hong Kong; Northwest University, China; Shandong University, China; Mahidol University, Thailand; and many more. His research has mostly revolved around class groups, Diophantine equations, elliptic curves, zeta values, and related topics, and he has published a considerable number of papers in respected journals. He has been involved in a number of conferences and received numerous national and international grants.
PREM PRAKASH PANDEY is Assistant Professor at the Indian Institute of Science Education and Research (IISER) Berhampur, Odisha. Before that, he was a postdoctoral fellow at HRI, Allahabad, and NISER Bhubaneswar, Odisha. After completing his Ph.D. at the Institute of Mathematical Sciences (IMSc), Chennai, he spent a couple of years at Chennai Mathematical Institute (CMI), Chennai, as a visiting scholar. Dr. Pandey's interests include class groups of number fields, annihilators of class groups, Diophantine equations, and related topics. During his time at HRI, he worked on divisibility problems for class numbers of quadratic fields with Dr. Hoque and Prof. Chakraborty.

Inhalt

1. A Geometric Approach to Large Class Groups: A Survey.- 2. On Simultaneous Divisibility of the Class Numbers of Imaginary Quadratic Fields.- 3. Thue Diophantine Equations: A Survey.- 4. A Lower Bound for the Class Number of Certain Real Quadratic Fields.- 5. A Survey of Certain Euclidean Number Fields.- Divisibility of Class Number of a Real Cubic or Quadratic Field and Its Fundamental Unit.- 6. Heights and Principal Ideals of Certain Cyclotomic Fields.- 7. Distribution of Residues Modulo p using the Dirichlet's Class Number Formula.- 8. On the Class Number Divisibility of Number Fields and Points on Elliptic Curves.- 9. Small Fields with Large Class Numbers.- 10. On the KummerVandiver Conjecture: An Extended Abstract.- 11. Cyclotomic Numbers and Jacobi Sums: A Survey.- 12. On LebesgueRamanujanNagell Type Equations.- 13. Partial Zeta Values and Class Numbers of R-D Type Real Quadratic Fields.- 14. A Pair of Quadratic Fields with Class Number Divisible by 3.