The Bloch–Kato Conjecture for the Riemann Zeta Function

John Coates, A. Raghuram, Anupam Saikia, R. Sujatha

• Publisher: Cambridge University Press
• Year: 2015
• Print ISBN: 9781107492967
• eBook ISBN: 9781316237526
• Topic: Mathematics
• Subtopic: Number Theory
• Index: Mathematics

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

Table of contents

• Cover
• Series information
• Title page
• Copyright information
• Table of contents
• List of contributors
• Preface
• 1 Special Values of the Riemann Zeta Function: Some Results and Conjectures
• 2 K-theoretic Background
• 3 Values of the Riemann Zeta Function at the Odd Positive Integers and Iwasawa Theory
• 4 Explicit Reciprocity Law of Bloch–Kato and Exponential Maps
• 5 The Norm Residue Theorem and the Quillen-Lichtenbaum Conjecture
• 6 Regulators and Zeta-functions
• 7 Soulé’s Theorem
• 8 Soulé’s Regulator Map
• 9 On the Determinantal Approach to the Tamagawa Number Conjecture
• 10 Motivic Polylogarithm and Related Classes
• 11 The Comparison Theorem for the Soulé–Deligne Classes
• 12 Eisenstein Classes, Elliptic Soulé Elements and the ell–Adic Elliptic Polylogarithm
• 13 Postscript