Zeta-functions of schemes in dimension 1 and 2

For schemes, there are two approaches to study of their zeta-functions, cohomological or motivic approach and adelic one. We will study the adelic approach for the zeta -functions of schemes which are algebraic curves or surfaces defined over a finite field. Firstly, we consider a version of the classical Tate-Iwasawa method for curves where we remove the well known manipulations with formulas and replace them by functoriality and duality considerations. Next, we discuss a possible extension of these constructions on the case of a surface.