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Published 2018 | Published + Submitted
Journal Article Open

Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

Abstract

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme X at any integer n in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum [65] and Geisser [36] for X of characteristic p, of Lichtenbaum [66] for X = Spec(O_F) and n = 0 where F is a number field, and of the second author for arbitrary X and n = 0 [72]. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou [31] if X is smooth over Spec(O_F), and hence that it holds in cases where the Tamagawa number conjecture is known.

Additional Information

© 2018 Documenta Mathematica. Attribution 4.0 International (CC BY 4.0).

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Submitted - 1605.01277.pdf

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Created:
August 19, 2023
Modified:
October 20, 2023