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13 - Postscript

Published online by Cambridge University Press:  05 March 2015

R. Sujatha
Affiliation:
University of British Columbia, Vancouver, Canada
John Coates
Affiliation:
University of Cambridge
A. Raghuram
Affiliation:
Indian Institute of Science Education and Research, Pune
Anupam Saikia
Affiliation:
Indian Institute of Technology, Guwahati
R. Sujatha
Affiliation:
University of British Columbia, Vancouver
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Summary

Abstract

The aim of this postscript is to provide a Leitfaden between the articles in this volume and their interlinkages, thereby clearly delineating the results from Galois cohomology and K-theory that are used in proving the main results in [BK90]. One reason to do this is to explicitly spell out the K-theoretic results that are used in [BK90], especially those of Soulé. We shall also indicate a proof of the finiteness of the global Tate–Shafarevich groups as considered in [BK90, 5.13] for M = ℤ(m); and note its relation to the Tate–Shafarevich groups considered by Fontaine and Perrin-Riou [FP94], as well as the Tate–Shafarevich groups that can be defined from the Poitou–Tate sequence. For simplicity, we only consider the base field ℚ, indicating briey how the results generalize to an arbitrary totally real abelian number field. As in the previous articles, p will denote an odd prime. All other notation is as in [Su15]. Nguyen Quang Do's contribution in putting this note together is gratefully acknowledged.

Leitfaden

In the article [Li15], Lichtenbaum used K-theory and étale cohomology along with the Chern class maps to prove Soulé's result on the vanishing of the cohomology groups for all m ≥ 2. Soulé [So79] proved that these groups are torsion and Schneider [Sc75] proved that they are divisible. The Quillen–Lichtenbaum conjecture asserts that the Chern class maps

are isomorphisms for p odd, k = 1, 2 and m ≥ 2. Soulé in [So79, Théoréme 6 (iii)] proved the surjectivity for even positive integers m. Dwyer and Friedlander [DF85] proved it for all integers m with k = 1 or 2 or 2mk > 1, using étale K-theory. The reader should also see [Ka93]. The vanishing of for all m ≥ 2 is equivalent to the finiteness of for m ≥ 2 and this will play an important role in the proof of the Bloch–Kato conjecture, as explained in the article by Coates [Co15] in this volume.

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Publisher: Cambridge University Press
Print publication year: 2015

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References

[BK90] Bloch, S., and Kato, K. 1990. L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, vol. 1. Progress in Math., 86, 333–400. Birkhäuser, Boston, MA.
[BN02] Benois, D., and Nguyen Quang Do, T. 2002. Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs ℚ(m) sur un corps abélien. Ann. Scient.Éc. Norm. Sup. 4e sér. t, 35, 641–672.Google Scholar
[Bl15] Blasius, D. 2015. Motivic Polylogarithm and related classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 193–209.
[Co15] Coates, J. 2015. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 45–64.
[DF85] Dwyer, W. G. and Friedlander, E. M. 1985. Algebraic and étale K-theory. Trans. AMS, 292, 247–280.Google Scholar
[FP94] Fontaine, J.-M., and Perrin-Riou, B. 1994. Autour des conjectures de Bloch et Kato, Cohomologie Galoisienne et valeurs de fonctions L, in “Motives”. Proc. Symp. in Pure Math., 55, 599–706.Google Scholar
[HK03] Huber, A., and Kings, G. 2003. Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J., 119, 3, 353–464.Google Scholar
[Hu15] Huber, A. 2015. A Motivic Construction of the Soulé Deligne Classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 210–238.
[KNF96] Kolster, M., Nguyen Quang Do, T. and Fleckinger, V. 1996. Twisted S-units, p-adic class number formulas and the Lichtenbaum conjectures. Duke Math. J., 84, 679–717.Google Scholar
[Ka93] Kahn, B. 1993. On the Lichtenbaum-Quillen conjecture, Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, 147–166. Kluwer Acad, Pub., Dordrecht.
[Ki15] Kings, G. 2015. The l-adic realisation of the elliptic polylogarithm and the evaluation of Eisenstein classes, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 239–296.
[Ko15] Kolster, M. 2015. The Norm residue homomorphism and the Quillen-Lichtenbaum conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 97–120.
[Li15] Lichtenbaum, S. 2015. Soulé's theorem, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 130–139.
[Ng15] Nguyen Quang Do, T. 2015. On the determinantal approach to the Tamagawa Number Conjecture, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 154–192.
[Ra15] Raghuram, A. 2015. Special values of the Riemann zeta function: some results and conjectures, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 1–21.
[Sa15] Saikia, A. 2015. Explicit reciprocity law of Bloch-Kato and exponential maps, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 65–96.
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[Su15] Sujatha, R. 2015. K-theoretic background, in The Bloch-Kato Conjecture for the Riemann Zeta Function, LMS Lecture Note Series 418. Cambridge University Press. 22–44.

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  • Postscript
    • By R. Sujatha, University of British Columbia, Vancouver, Canada
  • Edited by John Coates, University of Cambridge, A. Raghuram, Indian Institute of Science Education and Research, Pune, Anupam Saikia, Indian Institute of Technology, Guwahati, R. Sujatha, University of British Columbia, Vancouver
  • Book: The Bloch–Kato Conjecture for the Riemann Zeta Function
  • Online publication: 05 March 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316163757.014
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  • Postscript
    • By R. Sujatha, University of British Columbia, Vancouver, Canada
  • Edited by John Coates, University of Cambridge, A. Raghuram, Indian Institute of Science Education and Research, Pune, Anupam Saikia, Indian Institute of Technology, Guwahati, R. Sujatha, University of British Columbia, Vancouver
  • Book: The Bloch–Kato Conjecture for the Riemann Zeta Function
  • Online publication: 05 March 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316163757.014
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Postscript
    • By R. Sujatha, University of British Columbia, Vancouver, Canada
  • Edited by John Coates, University of Cambridge, A. Raghuram, Indian Institute of Science Education and Research, Pune, Anupam Saikia, Indian Institute of Technology, Guwahati, R. Sujatha, University of British Columbia, Vancouver
  • Book: The Bloch–Kato Conjecture for the Riemann Zeta Function
  • Online publication: 05 March 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781316163757.014
Available formats
×