IMO, even with 0 nitrates, you will have some algae on the tank
20 is NOT abad number for trates either, but just for kicks, lets start with which tester you use? and have you had it tested by someone else witha different kit(different type preferred)?
Every SW tank I have ever seen, has a similar problem with algae, it is par for the course. And yes, all those tanks in books and magazines are normally either just recently set up, or freshly cleaned
I do also agree, the lights will have no affect on nitrates, DIRECTLY. However, they will affect coraline and other algaes which cna help with nitrate control