2021/11/26 by Alexey Beshenov, Beshenov, Alexey
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2111.13398
openalex publication_date 2021/11/26 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 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 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-étale 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].