| 概要 |
Let X be an arithmetic scheme (i.e., separated, of finite type over Spec Z) of Krull dimension one. For the associated zeta-function ζ(X, s), we write down a formula for the special value at s = n < 0... in terms of the étale motivic cohomology of X and a regulator. We prove it in the case when, for each generic point η ∈ X with char κ(η) = 0, the extension κ(η)/Q is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-étale formalism developed by the author (arXiv preprints 2012.11034 and 2102.12114), following the work of Flach and Morin (Doc. Math. 23 (2018), 1425–1560). We also calculate the Weil-étale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from arXiv preprint 2102.12114.続きを見る
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Mendeley出力
