The Bloch-Kato conjecture on special values of L -functions. A survey of known results

Guido Kings

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 179-198
  • ISSN: 1246-7405

Abstract

top
This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.

How to cite

top

Kings, Guido. "The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results." Journal de théorie des nombres de Bordeaux 15.1 (2003): 179-198. <http://eudml.org/doc/249112>.

@article{Kings2003,
abstract = {This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.},
author = {Kings, Guido},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Bloch-Kato conjecture; motives; -functions; survey},
language = {eng},
number = {1},
pages = {179-198},
publisher = {Université Bordeaux I},
title = {The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results},
url = {http://eudml.org/doc/249112},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Kings, Guido
TI - The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 179
EP - 198
AB - This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.
LA - eng
KW - Bloch-Kato conjecture; motives; -functions; survey
UR - http://eudml.org/doc/249112
ER -

References

top
  1. [1] F. Bars, On the Tamagawa number conjecture for CM elliptic curves defined over Q. J. Number Theory95 (2002), 190-208. Zbl1081.11045MR1924097
  2. [2] A. Beilinson, Higher regulators and values of L-functions. Jour. Soviet. Math.30 (1985), 2036-2070. Zbl0588.14013
  3. [3] J.-R. Belliard, T. Nguyen QuangDo, Formules de classes pour les corps abéliens réels. Ann. Inst. Fourier (Grenoble) 51 (2001), 907-937. Zbl1007.11063MR1849210
  4. [4] D. Benois, T. Nguyen Quang Do, La conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien. Ann. Sci. Ecole Norm. Sup.35 (2002), 641-672. Zbl1125.11351MR1951439
  5. [5] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, 333-400, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990. Zbl0768.14001MR1086888
  6. [6] D. Burns, C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, preprint, 2000. Zbl1142.11076
  7. [7] D. Burns, M. Flach, Tamagawa numbers for motives with non-commutative coefficients. Doc. Math.6 (2001), 501-570. Zbl1052.11077MR1884523
  8. [8] P. Deligne, Valeurs de fonctions L et périodes d'intégrales. Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 313-346, Amer. Math. Soc., Providence, R.I., 1979. Zbl0449.10022MR546622
  9. [9] C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields I. Invent. Math.96 (1989), 1-69. Zbl0721.14004MR981737
  10. [10] C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields II. Ann. Math.132 (1990), 131-158. Zbl0721.14005MR1059937
  11. [11] F. Diamond, M. Flach, L. Guo, Adjoint motives of modular forms and the Tamagawa number conjecture, preprint, 2001. Zbl1121.11045
  12. [12] H. Esnault, On the Loday symbol in the Deligne-Beilinson cohomology. K-theory3 (1989), 1-28. Zbl0697.14006MR1014822
  13. [13] J.-M. Fontaine, Valeurs spéciales des fonctions L des motifs. Séminaire Bourbaki, Vol. 1991/92. Astérisque No. 206, (1992), Exp. No. 751, 4, 205-249. Zbl0799.14006MR1206069
  14. [14] J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato, cohomologie galoisienne et valeurs de fonctions L. Motives (Seattle, WA, 1991), 599-706, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. Zbl0821.14013MR1265546
  15. [15] C. Goldstein, N. Schappacher, Conjecture de Deligne et Γ-hypothese de Lichtenbaum sur les corps quadratiques imaginaires. C. R. Acad. Sci. Paris Sér. I Math.296 (1983), 615-618. Zbl0553.12003
  16. [16] L. Guo, On the Bloch-Kato conjecture for Hecke L-functions. J. Number Theory57 (1996), 340-365. Zbl0869.11055MR1382756
  17. [17] M. Harrison, On the conjecture of Bloch-Kato for Grössencharacters over Q(i). Ph.D. Thesis, Cambridge University, 1993. 
  18. [18] A. Huber, G. Kings, Degeneration of l-adic Eisenstein classes and of the elliptic poylog. Invent. Math.135 (1999), 545-594. Zbl0955.11027MR1669288
  19. [19] A. Huber, G. Kings, Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, to appear in Duke Math. J. Zbl1044.11095MR2002643
  20. [20] A. Huber, J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne. Doc. Math. J. DMV3 (1998), 27-133. Zbl0906.19004MR1643974
  21. [21] U. Jannsen, Deligne homology, Hodge-D-conjecture, and motives. In: Rapoport et al (eds.): Beilinson's conjectures on special values of L-functions. Academic Press, 1988. Zbl0701.14019MR944998
  22. [22] U. Jannsen, On the l-adic cohomology of varieties over number fields and its Galois cohomology. In: Ihara et al. (eds.): Galois groups over Q, MSRI Publication, 1989. Zbl0703.14010MR1012170
  23. [23] K. Kato, Iwasawa theory and p-adic Hodge theory. Kodai Math. J.16 (1993), no. 1, 1-31. Zbl0798.11050MR1207986
  24. [24] K. Kato, Lectures on the approach to Iwasawa theory for Hasse- Weil L-functions via BdR I. Arithmetic algebraic geometry (Trento, 1991), 50-163, Lecture Notes in Math.1553, Springer, Berlin, 1993. Zbl0815.11051MR1338860
  25. [25] K. Kato, Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions vis BdR, Part II, unpublished preprint, 1993. MR1338860
  26. [26] K. Kato, Euler systems, Iwasawa theory, and Selmer groups. Kodai Math. J.22 (1999), 313-372. Zbl0993.11033MR1727298
  27. [27] G. Kings, The Tamagawa number conjecture for CM elliptic curves. Invent. math.143 (2001), 571-627. Zbl1159.11311MR1817645
  28. [28] G. Kings, Higher regulators, Hilbert modular surfaces and special values of L-functions. Duke Math. J.92 (1998), 61-127. Zbl0962.11024MR1609325
  29. [29] M. Kolster, TH. Nguyen Quang Do, Universal distribution lattices for abelian number fields, preprint, 2000. 
  30. [30] M. Kolster, TH. Nguyen Quang Do, V. Fleckinger, Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J.84 (1996), no. 3, 679-717. Correction: Duke Math. J.90 (1997), no. 3, 641-643. Zbl0863.19003MR1408541
  31. [31] K. Kunnemann, On the Chow motive of an Abelian Scheme. Motives (Seattle, WA, 1991), 189-205, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. Zbl0823.14032MR1265530
  32. [32] Y.I. Manin, Correspondences, motives and monoidal transformations. Mat. Sbor.77, AMS Transl. (1970), 475-507. Zbl0199.24803
  33. [33] B. Mazur, A. Wiles, Class fields of abelian extensions of Q. Invent. Math.76 (1984), 179-330. Zbl0545.12005MR742853
  34. [34] J. Nekovar, Beilinson's conjectures. Motives (Seattle, WA, 1991), 537-570, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. Zbl0799.19003MR1265544
  35. [35] J. Neukirch, The Beilinson conjecture for algebraic number fields, in: M. Rapoport et al. (eds.): Beilinson's conjectures on Special Values of L-functions, Academic Press, 1988. Zbl0651.12009MR944995
  36. [36] K. Rubin, Euler systems. Annals of Mathematics Studies, 147, Princeton University Press, Princeton, NJ, 2000. Zbl0977.11001MR1749177
  37. [37] A.J. Scholl, Motives for modular forms. Invent. math.100 (1990), 419-430. Zbl0760.14002MR1047142
  38. [38] A. Wiles, The Iwasawa main conjecture for totally real fields. Ann. Math.131 (1990), 493-540. Zbl0719.11071MR1053488

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.