The Bloch–Kato Conjecture for the Riemann Zeta Function London Mathematical Society Lecture Note Series

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.

List of contributors; Preface A. Raghuram; 1. Special values of the Riemann zeta function: some results and conjectures A. Raghuram; 2. K-theoretic background R. Sujatha; 3. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory John Coates; 4. Explicit reciprocity law of Bloch–Kato and exponential maps Anupam Saikia; 5. The norm residue theorem and the Quillen–Lichtenbaum conjecture Manfred Kolster; 6. Regulators and zeta functions Stephen Lichtenbaum; 7. Soulé's theorem Stephen Lichtenbaum; 8. Soulé's regulator map Ralph Greenberg; 9. On the determinantal approach to the Tamagawa number conjecture T. Nguyen Quang Do; 10. Motivic polylogarithm and related classes Don Blasius; 11. The comparison theorem for the Soulé–Deligne classes Annette Huber; 12. Eisenstein classes, elliptic Soulé elements and the ℓ-adic elliptic polylogarithm Guido Kings; 13. Postscript R. Sujatha.

John Coates was Sadleirian Professor of Pure Mathematics at the University of Cambridge from 1986 until 2012. Most of his research has focused on the mysterious, and still largely conjectural, connections between special values of L-functions and purely arithmetic questions, largely via the p-adic techniques of Iwasawa theory. Professor Coates was elected a fellow of the Royal Society of London in 1985, and served as President of the London Mathematical Society from 1988 to 1990. He was awarded the Senior Whitehead Prize by the London Mathematical Society in 1997.
A. Raghuram is the Coordinator for Mathematics at the Indian Institute of Science Education and Research (IISER) at Pune. Previously, he was a tenured faculty member of the Department of Mathematics at Oklahoma State University. He has also held various visiting positions at the University of Iowa, Purdue University, and the Max Planck Institute for Mathematics in Germany. Professor Raghuram's research interests concern the arithmetic properties of automorphic forms. He uses analytic methods in the Langlands Program and geometric tools from the cohomology of arithmetic groups to study the special values of L-functions. His research has been supported by the National Science Foundation, USA, and the Alexander von Humboldt Foundation, Germany.
Anupam Saikia is an Associate Professor in the Department of Mathematics at the Indian Institute of Technology Guwahati. Previously, he was a William Hodge Fellow at IHES, France, and a CRM-ISM postdoctoral fellow at McGill University after completing his PhD at DPMMS, University of Cambridge. The main theme of his research is Iwasawa theory of cyclotomic fields, elliptic curves and p-adic measures.
R. Sujatha is a Professor at the University of British Columbia, Vancouver. She conducts research in the broad area of algebraic number theory and has authored articles on the algebraic theory of quadratic forms, Iwasawa theory, and the study of motives. Professor