Let $X$ be an arithmetic scheme (i.e., separated, of finite type over $\operatorname{Spec} \mathbb{Z}$) of Krull dimension $1$. For the associated zeta function $��(X,s)$, we write down a formula for the special value at $s = n < 0$ in terms of the ��tale motivic cohomology of $X$ and a regulator. We prove it in the case when for each generic point $��\in X$ with $\operatorname{char} ��(��) = 0$, the extension $��(��)/\mathbb{Q}$ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-��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-��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].

29 pages, comments are welcome!