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Higher regulators, algebraic $K$-theory, and zeta functions of elliptic curves. (English) Zbl 0958.19001
CRM Monograph Series. 11. Providence, RI: American Mathematical Society (AMS). ix, 97 p. $ 24.00 (2000).
The editors were indeed well-advised to publish these lecture notes, which were given at Irvine in 1977. They remain one of the best introductions to the study of special values of $L$-functions in arithmetic geometry. In them, not only will the reader find the first general statement about the value at 2 of the $L$-function of an elliptic curve, which has influenced much of the subsequent work of Beilinson et al. (see e.g. the book edited by Rapoport, Schneider and Schappacher, based on the meeting of the Arbeitsgemeinschaft Geyer-Harder held in Oberwolfach, FRG in April 1986 [“Beilinson’s conjectures on special values of L-functions”, Perspect. Math. 4 (1988; Zbl 0635.00005)]), but he will also find a presentation of the work of Borel on the Dedekind zeta function. If it is true, as the author says in his apology, that this text compares to a Model T Ford, given that nowadays there are much faster cars to drive around the country of special values, it could also be said that often one is taken on trips aboard cars that drive so fast, that one cannot enjoy the beauty of the scenary. Bloch’s text offers a beautiful mixture of old and new, definitely turned towards the new. It shows for instance the interest for arithmetic of the study of higher K-theory (something already indicated by Lichtenbaum’s conjectures, but remember that Quillen’s seminal paper … dates of 1973; also Bloch deals with the K-theory of a scheme not a ring). The only regret we may express is that the editors did not add to the original text remarks on the work in the field opened by Bloch’s text in the last two decades. We would have mentioned {\it A. A. Beilinson}’s 1980 paper [J. Sov. Math. 30, 2036-2070 (1985; Zbl 0588.14013)], {\it S. Bloch}’s article on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture [Invent. Math. 58, 65-76 (1980; Zbl 0444.14015)], {\it S. Bloch} and {\it K. Kato}’s paper of 1990 [The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333-400 (1990; Zbl 0768.14001)], {\it D. Zagier}’s paper on polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields [Arithmetic algebraic geometry, Prog. Math. 89, 391-430 (1991; Zbl 0728.11062)], and, much in the style of Bloch’s lectures, the recent work of {\it D. W. Boyd} on Mahler measures and special values of $L$-functions [Exp. Math. 7, No. 1, 37-82 (1998; Zbl 0932.11069)]. At any rate the book under review should be (at least) in every library of every department interested in modern number theory.

19F27Étale cohomology, higher regulators, zeta and $L$-functions ($K$-theory)
14G10Zeta-functions and related questions
14-02Research monographs (algebraic geometry)
19D50Computations of higher $K$-theory of rings
19-02Research monographs ($K$-theory)
11G40$L$-functions of varieties over global fields