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Introduction to the Beilinson conjectures. (English) Zbl 0673.14007
Beilinson’s conjectures on special values of L-functions, Meet. Oberwolfach/FRG 1986, Perspect. Math. 4, 1-35 (1988).
[For the entire collection see Zbl 0635.00005.]
The leading coefficients at integral arguments of the L-functions of algebraic varieties over number fields seem to be closely related to the global arithmetical geometry of these varieties. A. A. Bejlinson [in J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)] has developed a general conjectural formalism which connects the “transcendental” parts of that leading coefficients to so-called regulators, defined by purely algebraic and geometrical means.
The author presents Beilinson conjectures in a way which leads the reader to their statement as directly as possible. A special attention is given to the construction of the Chern class maps from the higher algebraic K- theory into any reasonable cohomology theory, which is a basic ingredient in the definition of the Beilinson regulators.

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions
11R70 \(K\)-theory of global fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)