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

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Guido Kings
The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results
Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 179-198, doi: 10.5802/jtnb.396
Article PDF | Reviews MR 2019010 | Zbl 1050.11063 | 1 citation in Cedram

Résumé - 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.

Bibliography

[1] F. Bars, On the Tamagawa number conjecture for CM elliptic curves defined over Q. J. Number Theory 95 (2002), 190-208. MR 1924097 | Zbl 1081.11045
[2] A. Beilinson, Higher regulators and values of L-functions. Jour. Soviet. Math. 30 (1985), 2036-2070. Zbl 0588.14013
[3] J.-R. Belliard, T. Nguyen Quang Do, Formules de classes pour les corps abéliens réels. Ann. Inst. Fourier (Grenoble) 51 (2001), 907-937.
Cedram | MR 1849210 | Zbl 1007.11063
[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.
Numdam | MR 1951439 | Zbl 01910884
[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. MR 1086888 | Zbl 0768.14001
[6] D. Burns, C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, preprint, 2000.
[7] D. Burns, M. Flach, Tamagawa numbers for motives with non-commutative coefficients. Doc. Math. 6 (2001), 501-570. MR 1884523 | Zbl 1052.11077
[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. MR 546622 | Zbl 0449.10022
[9] C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields I. Invent. Math. 96 (1989), 1-69. MR 981737 | Zbl 0721.14004
[10] C. Deninger, Higher regulators and Hecke L-series of imaginary quadratic fields II. Ann. Math. 132 (1990), 131-158. MR 1059937 | Zbl 0721.14005
[11] F. Diamond, M. Flach, L. Guo, Adjoint motives of modular forms and the Tamagawa number conjecture, preprint, 2001.
[12] H. Esnault, On the Loday symbol in the Deligne-Beilinson cohomology. K-theory 3 (1989), 1-28. MR 1014822 | Zbl 0697.14006
[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.
Numdam | MR 1206069 | Zbl 0799.14006
[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. MR 1265546 | Zbl 0821.14013
[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. Zbl 0553.12003
[16] L. Guo, On the Bloch-Kato conjecture for Hecke L-functions. J. Number Theory 57 (1996), 340-365. MR 1382756 | Zbl 0869.11055
[17] M. Harrison, On the conjecture of Bloch-Kato for Grössencharacters over Q(i). Ph.D. Thesis, Cambridge University, 1993.
[18] A. Huber, G. Kings, Degeneration of l-adic Eisenstein classes and of the elliptic poylog. Invent. Math. 135 (1999), 545-594. MR 1669288 | Zbl 0955.11027
[19] A. Huber, G. Kings, Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, to appear in Duke Math. J. MR 2002643 | Zbl 1044.11095
[20] A. Huber, J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne. Doc. Math. J. DMV 3 (1998), 27-133. MR 1643974 | Zbl 0906.19004
[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. MR 944998 | Zbl 0701.14019
[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. MR 1012170 | Zbl 0703.14010
[23] K. Kato, Iwasawa theory and p-adic Hodge theory. Kodai Math. J. 16 (1993), no. 1, 1-31.
Article | MR 1207986 | Zbl 0798.11050
[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. MR 1338860 | Zbl 0815.11051
[25] K. Kato, Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions vis BdR, Part II, unpublished preprint, 1993. MR 1338860
[26] K. Kato, Euler systems, Iwasawa theory, and Selmer groups. Kodai Math. J. 22 (1999), 313-372.
Article | MR 1727298 | Zbl 0993.11033
[27] G. Kings, The Tamagawa number conjecture for CM elliptic curves. Invent. math. 143 (2001), 571-627. MR 1817645 | Zbl 01586347
[28] G. Kings, Higher regulators, Hilbert modular surfaces and special values of L-functions. Duke Math. J. 92 (1998), 61-127.
Article | MR 1609325 | Zbl 0962.11024
[29] M. Kolster, TH. Nguyen Quang Do, Universal distribution lattices for abelian number fields, preprint, 2000.
[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. MR 1408541
[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. MR 1265530 | Zbl 0823.14032
[32] Y.I. Manin, Correspondences, motives and monoidal transformations. Mat. Sbor. 77, AMS Transl. (1970), 475-507. Zbl 0199.24803
[33] B. Mazur, A. Wiles, Class fields of abelian extensions of Q. Invent. Math. 76 (1984), 179-330. MR 742853 | Zbl 0545.12005
[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. MR 1265544 | Zbl 0799.19003
[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. MR 944995 | Zbl 0651.12009
[36] K. Rubin, Euler systems. Annals of Mathematics Studies, 147, Princeton University Press, Princeton, NJ, 2000. MR 1749177 | Zbl 0977.11001
[37] A.J. Scholl, Motives for modular forms. Invent. math. 100 (1990), 419-430. MR 1047142 | Zbl 0760.14002
[38] A. Wiles, The Iwasawa main conjecture for totally real fields. Ann. Math. 131 (1990), 493-540. MR 1053488 | Zbl 0719.11071