Étale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

Abstract In this paper, we deal with the etale cohomology of a proper regular arithmetic scheme X with 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 X with 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.