Etale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

[摘要] In this paper, we deal with the etale cohomology of a proper regular arithmetic scheme Xwith Z(p)(r) and Q(p)(r)-coefficients, where the coefficients are complexes of etale sheaves that the author introduced in [SH]. We will prove that the etale cohomology of Xwith Q(p)(r)-coefficients agrees with the Selmer group of Bloch-Kato for any r >= dim(X). Using this fundamental result, we further discuss an approach to the study of zeta values (or residue) at s = r, via the etale cohomology with Z(p)(r)-coefficients, relating Tamagawa number conjecture of Bloch-Kato with a zeta value formula. As a consequence, we will obtain an unconditional example of an arithmetic surface for which the residue of its zeta function at s = 2 is computed modulo rational numbers prime to p, for infinitely many p's. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-01

[关键词] Etale cohomology;Motivic cohomology;Arithmetic schemes;Selmer groups;Tate-Shafarevich groups and zeta;values of arithmetic surfaces