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Values of zeta-functions at non-negative integers. (English) Zbl 0591.14014

Number theory, Proc. Journ. arith., Noordwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 127-138 (1984).
The author deals with the values of zeta-functions \(\zeta(X,s)\) of a schema of finite type over \(\operatorname{Spec}\mathbb{Z}\) at non-negative integral values of \(s\). For \(s=0\) the behaviour of the \(\zeta\)-function is closely related to the étale cohomology of the constant sheaf \(\mathbb{Z}\) and for \(s=1\) with that of the sheaf \(G_m\). It seems necessary for \(s\ge 2\) to consider certain complexes of sheaves. The author states “that the very existence of such complexes is at the moment hypothetical, but hypothetical properties of these hypothetical complexes present a fascinating picture, well worth investigation”. The reviewer does agree with this statement!
The author formalizes his research by the introduction of a list of “axioms” for the hypothetical complexes; these are of the type of generalizations of Hilbert 90, of the Merkur’ev-Suslin theorem and so on. Independently, A. A. Beĭlinson made analogous conjectures. The hypothetical complexes should give rise to a vast duality theorem, generalizing and clarifying simultaneously many seemingly unrelated results. This duality theorem generalizes many known duality-theorems in local and global class field theory.
In his last paragraph the author leads us to really deep waters in going over to \(\zeta\)-functions of algebraic number fields.
[For the entire collection see Zbl 0535.00008.]

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
11R42 Zeta functions and \(L\)-functions of number fields
11S31 Class field theory; \(p\)-adic formal groups

Citations:

Zbl 0535.00008
Full Text: DOI