Beilinson’s conjectures on special values of \(L\)-functions (based on the meeting of the Arbeitsgemeinschaft Geyer-Harder held in Oberwolfach, FRG in April 1986). (English) Zbl 0635.00005

Perspectives in Mathematics, 4. Boston, MA etc.: Academic Press, Inc. xxiii, 373 p.; $ 37.50 (1988).

The collection deals with Alexander Beilinson’s conjectures on special values of \(L\)-functions. Topics covered range from Pierre Deligne’s conjecture on critical values of \(L\)-functions to the Deligne-Beilinson cohomology, along with the Beilinson conjecture for algebraic number fields and Riemann-Roch theorem. Beilinson’s regulators are also compared with those of Émile Borel. Comprised of 10 chapters, this volume begins with an introduction to the Beilinson conjectures and the theory of Chern classes from higher \(K\)-theory. The “simplest” example of an \(L\)-function is presented, the Riemann zeta function. The discussion then turns to Deligne’s conjecture on critical values of \(L\)-functions and its connection to Beilinson’s version. Subsequent chapters focus on the Deligne-Beilinson cohomology, \(\lambda\)-rings and Adams operations in algebraic \(K\)-theory, Beilinson conjectures for elliptic curves with complex multiplication, and Beilinson’s theorem on modular curves. The book concludes by reviewing the definition and properties of Deligne homology, as well as Hodge-\(\mathcal D\)-conjecture. This monograph should be of considerable interest to researchers and graduate students who want to gain a better understanding of Beilinson’s conjectures on special values of \(L\)-functions.
The articles of this volume will be reviewed individually under the abbreviation “Beilinson’s conjectures on special values of \(L\)-functions, Meet. Oberwolfach/FRG 1986, Perspect. Math. 4”.

MSC:

00B25	Proceedings of conferences of miscellaneous specific interest
14-06	Proceedings, conferences, collections, etc. pertaining to algebraic geometry

Keywords:

Beilinson’s conjectures; L-functions; Meeting; Oberwolfach (Germany)