While in the case of affine curves over a finite field the Isom-conjecture was solved by Tamagawa, until this moment almost nothing is known about the case of arithmetic curves, mainly because of the lack of a satisfactory cohomology theory, which is extensively used in Tamagawa's approach (as the fundamental group of an arithmetic curve is not enough well understood). We present a new method to overcome at least the first difficulty, i.e., how to reconstruct the decomposition subgroups of points inside the fundamental group, using L-functions and the class number formula and omitting the use of any cohomology theory.