Does aluminium carbonate exist?
It does not exist in nature, but you could make it in the lab by adding pressure.

From a traditional pencil and paper calculation I found the pressure needed for 

Al2/3O (s) + CO2 (g) = Al2/3CO3 (s) 

is about 10&nbsp;GPa...

Or you could make an Al2(CO3)3 (aq) solution with concentration of $10^{-5}$ M.

Well, still not impossible. And there should be some special solvents and/or complex structures that easily allow this reaction (postulation).