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).

[For the entire collection see Zbl 0535.00008.]

The author deals with the values of zeta-functions \(\zeta\) (X,s) of a schema of finite type over 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 \(s\geq 2\) 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. Bejlinson 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.

The author deals with the values of zeta-functions \(\zeta\) (X,s) of a schema of finite type over 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 \(s\geq 2\) 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. Bejlinson 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.

Reviewer: F.van der Blij

##### 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 |