Étale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

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