Zeta-values of arithmetic schemes at negative integers and Weil-étale cohomology

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Zeta-values of arithmetic schemes at negative integers and Weil-étale cohomology

Type: Doctoral Thesis
Title: Zeta-values of arithmetic schemes at negative integers and Weil-étale cohomology
Author: Beshenov, A.
Issue Date: 2018-12-10
Keywords: Arithmetic geometry
Zeta functions
Weil-étale cohomology
Special values
Motivic cohomology
Abstract: This work is dedicated to interpreting in cohomological terms the special values of zeta functions of arithmetic schemes. Baptiste Morin and Matthias Flach gave a construction of Weil-étale cohomology using Bloch's cycle complexes and stated a precise conjecture for the special values of proper regular arithmetic schemes at any integer argument s=n. The goal of this thesis is to generalize their constructions to arbitrary arithmetic schemes (possibly singular or non-proper), while restricting to the case n We prove that the resulting conjecture is compatible with the decomposition of an arbitrary scheme into an open subscheme and its closed complement. We also show that this conjecture for an arithmetic scheme X at s=n is equivalent to the conjecture for A^r_X at s=n-r, for any r >= 0. It follows that, taking as an input the schemes for which the conjecture is known, it is possible to construct new schemes, possibly singular or non-proper, for which the conjecture holds as well. This is the main unconditional outcome of the machinery developed in this thesis.
Promotor: Supervisor: Edixhoven S.J., Morin B.
Faculty: Faculty of Science
University: Leiden University and Bordeaux University
Handle: http://hdl.handle.net/1887/68224
 

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