# Special Values of Zeta Functions of Schemes

@article{Lichtenbaum2017SpecialVO, title={Special Values of Zeta Functions of Schemes}, author={Stephen Lichtenbaum}, journal={arXiv: Algebraic Geometry}, year={2017} }

Let X be a regular scheme, projective and flat over Spec \mathbb Z. We give two conjectural formulas, up to sign and powers of 2, for \zeta^*(X,r), the leading term in the series expansion of \zeta(X,s) at s=r. The first formula builds on work of Fontaine and Perrin-Riou replacing \mathbb Q_structuers by \mathbb Z-structures, and the second is a very simple formula in terms of the Euler characteristic of a certain complex of sheaves in a hypothetical Weil-etale Grothendieck topology. A…

## 4 Citations

### Weil-\'etale cohomology and zeta-values of arithmetic schemes at negative integers

- Mathematics
- 2021

Following the ideas of Flach and Morin [11], we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function ζ(X, s) at s = n < 0, where X is a…

### Weil-étale cohomology for arbitrary arithmetic schemes and n < 0 . Part II : The special value conjecture

- Mathematics
- 2021

Following the ideas of Flach and Morin [FM2018], we state a conjecture in terms of Weil-étale cohomology for the vanishing order and special value of the zeta function ζ(X, s) at s = n < 0, where X…

### Zeta-values of one-dimensional arithmetic schemes at strictly negative integers

- Mathematics
- 2021

Let X be an arithmetic scheme (i.e., separated, of finite type over SpecZ) 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…

### Weil-etale cohomology and special values of L-functions at zero

- Mathematics
- 2015

We construct the Weil-\'etale cohomology and Euler characteristics for a subclass of the class of $\mathbb{Z}$-constructible sheaves on the spectrum of the ring of integers of a totally imaginary…

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