Suggested answer:
(1) Examples of authority on and authority over:
I daresay I'm actually the greatest living authority on Vangmoor.
A constable has no authority over me.
(2) Authority on has the meaning of "expert", i.e. if you are an authority on something, you are an expert on it. Authority over, on the other hand, has the meaning of "power", i.e. if you have authority over something, you have power over it.
(3) Another combination of authority + preposition with the same meaning as authority on is authority upon, e.g.:
...he was a recognized authority upon the subject...
Read about noun + preposition combinations here
Terminology: preposition