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On the Work of Ritter and Weiss in Comparison with Kakde’s Approach

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Book cover Noncommutative Iwasawa Main Conjectures over Totally Real Fields

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 29))

Abstract

In this chapter a survey of the proof by Ritter and Weiss of the Iwasawa Main Conjecture over totally real fields for the Tate motive in comparison with Kakde’s approach is provided. In particular, we shall compare the two different descriptions of \({K}_{1}(\Lambda (G))\): one explicitly given in Kakde’s work as presented in the contribution Schneider and Venjakob [SV] of this volume, the other one derived from Ritter and Weiss’ work; the latter description given in Sect. 6 is new.

MSCs: 11R23, 19B28, 11S23, 20C10, 11S40

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Notes

  1. 1.

    In order to remedy this Ritter and Weiss are apparently thinking about writing a lecture note volume about their approach.

  2. 2.

    As M.  Kakde kindly pointed out to me it seems quite difficult to check directly that the set satisfying the conditions below forms a subgroup, so this becomes only clear a posteriori once it is shown that this set is the full image of K 1. 

References

  1. R. Boltje, A canonical Brauer induction formula. Astérisque 181–182(5), 31–59 (1990)

    Google Scholar 

  2. D. Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, Preprint (2011)

    Google Scholar 

  3. J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The GL 2 main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES. 101(1), 163–208 (2005)

    Google Scholar 

  4. T. Fukaya, K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proceedings of the St. Petersburg Mathematical Society, vol. XII (AMS, Providence, 2006), pp. 1–86 (Amer. Math. Soc. Transl. Ser. 2, vol. 219)

    Google Scholar 

  5. T. Hara, Iwasawa theory of totally real fields for certain non-commutative p-extensions. J. Number Theory 130(4), 1068–1097 (2010)

    Google Scholar 

  6. M. Kakde, Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases. J Algebr. Geom. 20, 631–683 (2011)

    Google Scholar 

  7. M. Kakde, Integral logarithm for completions of a localisation of Iwasawa algebras, preprint (2011)

    Google Scholar 

  8. M. Kakde, The main conjecture of Iwasawa theory for totally real fields, preprint (2011)

    Google Scholar 

  9. M. Kakde, Congruences between abelian p-adic zeta functions, this volume (2012)

    Google Scholar 

  10. K. Kato, Iwasawa theory for totally real fields for Galois extensions of Heisenberg type, preprint (‘Very preliminary version’) (2007)

    Google Scholar 

  11. J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields. Grundlehren der mathematischen Wissenschaften, vol. 323 (Springer, Berlin/New York, 2000)

    Google Scholar 

  12. A. Nickel, Equivariant Iwasawa theory and non-abelian Stark-type conjectures, Preprint, arXiv:1109.5525 (2011)

    Google Scholar 

  13. R. Oliver, Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge, 1988)

    Google Scholar 

  14. J. Ritter, A. Weiss, The lifted root number conjecture and Iwasawa theory. Mem. Am. Math. Soc. 157(748), viii+90 (2002)

    Google Scholar 

  15. J. Ritter, A. Weiss, Toward equivariant Iwasawa theory. Manuscr. Math. 109(2), 131–146 (2002)

    Google Scholar 

  16. J. Ritter, A. Weiss, Toward equivariant Iwasawa theory II. Indag. Math. (N.S.) 15(4), 549–572 (2004)

    Google Scholar 

  17. J. Ritter, A. Weiss, Toward equivariant Iwasawa theory IV. Homol. Homotopy Appl. 7(3), 155–171 (2005)

    Google Scholar 

  18. J. Ritter, A. Weiss, Toward equivariant Iwasawa theory III. Math. Ann. 336(1), 27–49 (2006)

    Google Scholar 

  19. J. Ritter, A. Weiss, Congruences between abelian pseudomeasures. Math. Res. Lett. 15(4), 715–725 (2008)

    Google Scholar 

  20. J. Ritter, A. Weiss, Equivariant Iwasawa theory: an example. Doc. Math. 13, 117–129 (2008)

    Google Scholar 

  21. J. Ritter, A. Weiss, Non-abelian pseudomeasures and congruences between abelian Iwasawa L-functions. Pure Appl. Math. Q. 4(4, Special Issue: In honor of Jean-Pierre Serre. Part 1), 1085–1106 (2008)

    Google Scholar 

  22. J. Ritter, A. Weiss, The integral logarithm in Iwasawa theory: an exercise. J. Théor. Nombres Bordeaux 22(1), 197–207 (2010)

    Google Scholar 

  23. J. Ritter, A. Weiss, On the ‘main conjecture’ of equivariant Iwasawa theory. J. Am. Math. Soc. 24, 1015–1050 (2011)

    Google Scholar 

  24. P. Schneider, O. Venjakob, K 1 of certain Iwasawa algebras – after Kakde, this volume (2012)

    Google Scholar 

  25. V.P. Snaith, Explicit Brauer induction. Cambridge Studies in Advanced Mathematics, vol. 40 (Cambridge University Press, Cambridge, 1994). With applications to algebra and number theory

    Google Scholar 

  26. R. Sujatha, Reductions of the main conjecture, this volume (2012)

    Google Scholar 

  27. C.A. Weibel, An introduction to algebraic K-theory (2011), http://math.rutgers.edu:80/~textasciitildeweibel/Kbook.html, in progress

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Authors and Affiliations

  1. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120, Heidelberg, Germany

    Otmar Venjakob

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  1. Otmar Venjakob
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Correspondence to Otmar Venjakob .

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Editors and Affiliations

  1. Department of Pure Mathematics, and Mathematical Statistics (DPMMS), University Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

    John Coates

  2. Institute of Mathematics, Westphalian University of Münster, Einsteinstrasse 62, Münster, 48149, Germany

    Peter Schneider

  3. Department of Mathematics, University of British Columbia, Mathematics Road 1984, Vancouver, V6T 1Z2, British Columbia, Canada

    R. Sujatha

  4. Institute of Mathematics, University of Heidelberg, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Otmar Venjakob

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Venjakob, O. (2013). On the Work of Ritter and Weiss in Comparison with Kakde’s Approach. In: Coates, J., Schneider, P., Sujatha, R., Venjakob, O. (eds) Noncommutative Iwasawa Main Conjectures over Totally Real Fields. Springer Proceedings in Mathematics & Statistics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32199-3_6

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