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The Bloch-Kato conjecture on special values of \(L\)-functions. A survey of known results. (English) Zbl 1050.11063

The author gives a survey of known results about a conjecture of S. Bloch and K. Kato, first formulated in “\(L\)-functions and Tamagawa numbers of motives” [The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)], which conjecturally links special values of \(L\)-functions of motives to cohomological data. In the number field case the conjecture (and more generally the equivariant version of the conjecture) is known for Abelian fields up to \(2\)-primary information. The most general result in this direction is contained in [D. Burns and C. Greither, Invent. Math. 153, 303–359 (2003; Zbl 1142.11076)].
The other cases discussed in the paper are certain elliptic curves with complex multiplication [cf. G. Kings, Invent. Math. 143, 571–627 (2001; Zbl 1159.11311)] and adjoint motives of newforms of weight \(\geq 2\) [cf. F. Diamond, M. Flach and L. Guo, Math. Res. Lett. 8, No. 4, 437–442 (2001; Zbl 1022.11023)].

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11-02 Research exposition (monographs, survey articles) pertaining to number theory
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R23 Iwasawa theory
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References:

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