Etale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

JOURNAL OF NUMBER THEORY

卷:227

Article

Sato, Kanetomo¹ 

[1] Chuo Univ, Dept Math, Bunkyo Ku, 1-13-27 Kasuga, Tokyo 1128551, Japan

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

DOI  :  10.1016/j.jnt.2021.03.020

来源: Elsevier

【 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.

【 授权许可】

Free