Papers › Zeta-values of one-dimensional arithmetic schemes at strictly negative integers
Zeta-values of one-dimensional arithmetic schemes at strictly negative integers
Alexey Beshenov
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Let X be an arithmetic scheme (i.e., separated, of finite type over Spec ℤ) of Krull dimension $1$. 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 \'{e}tale 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-\'{e}tale formalism developed by the author in [arXiv:2012.11034] and [arXiv:2102.12114], following the work of Flach and Morin (Doc. Math. 23 (2018), 1425--1560). We also calculate the Weil-\'{e}tale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from [arXiv:2102.12114].
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