The Bloch-Kato Conjecture for the Riemann Zeta Function

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. Soule's theorem Stephen Lichtenbaum; 8. Soule'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 Soule-Deligne classes Annette Huber; 12. Eisenstein classes, elliptic Soule elements and the l-adic elliptic polylogarithm Guido Kings; 13. Postscript R. Sujatha.