Compatibility of Special Value Conjectures with the Functional Equation of Zeta Functions
We prove that the special value conjecture for the Zeta function ζ(X, s) of a proper, regular arithmetic scheme X that we formulated in  is compatible with the functional equation of ζ(X, s) provided that the rational factor C(X, n) we were not able to compute previously has the simple explicit form given in the introduction below.
Additional InformationCC BY 4.0. We would like to thank S. Lichtenbaum for many indirect contributions to this project. Our realization that his Conjecture 0.1 in  could be proven using the ideas of T. Saito in  was at the origin of this article (but in fact such a proof had already been carried out by T. Saito himself in [Cor. 4.9], see Thm. 3.3 below). Lichtenbaum's preprint  has considerable overlap with our article in that he also formulates a conjecture on special values of ζ(X, s) and proves compatibility with the functional equation. Despite differences in language, and the fact that all results of  are only up to powers of 2, we believe our approaches are largely equivalent. The first version of  was posted in April 2017 and the authors recall discussing an explicit formula for C(X, n) among each other at around the same time. However, to the best of our knowledge we never communicated with Lichtenbaum about specifics of special value conjectures, and Lichtenbaum and us arrived at our respective formulations independently. We would also like to thank Spencer Bloch for interesting discussions related to C(X, n). The second author was supported by ANR-15-CE40-0002-01. The first author was supported by collaboration grant 522885 from the Simons foundation.
Published - 10012165000.pdf
Submitted - 2005.04829.pdf