\(L\)-functions at the central critical point. (English) Zbl 0807.14015

Gross, Benedict H.

Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 527-535 (1994).

The of this paper is about the special values of \(L\)-functions related to Deligne’s general conjecture [P. Deligne in: Automorphic forms, representations and \(L\)-functions, Proc. Symp. Pure Math. 33, No. 2, 313- 346 (1979; Zbl 0449.10022)]. The author begins with a brief review on Deligne’s conjecture and then describes an extension of the conjecture by A. Beilinson [in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata 1985, Contemp. Math. 67, 1-24 (1987; Zbl 0624.14005) and S. Bloch [J. Pure Appl. Algebra 34, 119- 145 (1984; Zbl 0577.14004)] of the leading term in the Taylor series of the \(L\)-function at the center of its critical strip. Finally, the author proves some statement about symplectic local root numbers, which play an important role in the study of central critical behavior.
For the entire collection see [Zbl 0788.00053].

Reviewer: M.Muro (Yanagido)

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

C-M field; abelian variety; center of critical strip of \(L\)-functions; special values of \(L\)-functions; Deligne’s general conjecture

Citations:

Zbl 0449.10022; Zbl 0624.14005; Zbl 0577.14004