TITLE: why Riemann Hypothesis over curves is easy but 'normal' hypothesis for the Riemann Zeta function $ \zeta (s) $ is hard , 
QUESTION [6 upvotes]: despite both zetas $ \zeta (s,X) $ and $ \zeta (s)$ have the same functional equation, the same Euler prodcut and the same Riemann-Weil formula 
why one of them is 'easy' and can be solved but the other is so hard?? the zeta one $ \zeta (s)$

REPLY [3 votes]: An additional issue is that the adelic norm map in the function field has discrete image, so its Pontryagin dual is actually compact. In the case of a number field, it has non-discrete, non-compact image, so its Pontryagin dual is non-compact as well. Since Tate's thesis, we know that the Fourier analysis on these duals is intimately related to $L$-functions. 
After-thought:
I guess this boils down to the same issue, which GH already mentioned: we have not only one prime field for all residue fields, for number fields there are infinitely many. In fact, there is no residue field for archimedean fields (yet).