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Zbl 1050.11063
Kings, Guido
The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results. (English)
[J] J. Théor. Nombres Bordx. 15, No. 1, 179-198 (2003). ISSN 1246-7405

The author gives a survey of known results about a conjecture of {\it S. Bloch} and {\it 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 [{\it D. Burns} and {\it C. Greither}, Invent. Math. 153, 303--359 (2003; Zbl 1142.11076)]. \par The other cases discussed in the paper are certain elliptic curves with complex multiplication [cf. {\it G. Kings}, Invent. Math. 143, 571--627 (2001; Zbl 1159.11311)] and adjoint motives of newforms of weight $\geq 2$ [cf. {\it F. Diamond, M. Flach} and {\it L. Guo}, Math. Res. Lett. 8, No. 4, 437--442 (2001; Zbl 1022.11023)].
[Manfred Kolster (Hamilton/Ontario)]

MSC 2000:: *11G40 L-functions of varieties over global fields
11-02 Research monographs (number theory)
14G10 Zeta-functions and related questions
11R23 Iwasawa theory

Keywords: Bloch-Kato conjecture; motives; $L$-functions; survey

Citations: Zbl 0768.14001; Zbl 1022.11023; Zbl 1142.11076; Zbl 1159.11311

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