Special Values of Zeta Functions of Schemes

2017/03/31 by Stephen Lichtenbaum, Lichtenbaum, Stephen
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #14G10(Primary) 11G40 14F20 14F42 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1704.00062

openalex publication_date 2017/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a regular scheme, projective and flat over Spec \mathbb Z. We give a conjectural formula, up to sign and powers of 2, for ζ^*(X,r), the leading term in the series expansion of ζ(X,s) at s=r, in terms of Weil-etale motivic cohomology, singular cohomology, and derived de Rham cohomology. This formula builds on work of Fontaine and Perrin-Riou replacing \mathbb Q_structuers by \mathbb Z-structures. We show that our conjectured formula is compatible with Serre's functional equation for the zeta function of X.

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