Étale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

Abstract

In this paper, we deal with the étale cohomology of a proper regular arithmetic scheme X with Zp(r) and Qp(r)-coefficients, where the coefficients are complexes of étale sheaves that the author introduced in [SH]. We will prove that the étale cohomology of X with Qp(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 étale cohomology with Zp(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.