## 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

Zbl 0535.00008
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