Can they turn me down??

So I have an emotional support animal. I have papers from a doctor and everything. I was told that if I rent from this place that I am not allowed to have her. Can they legally do that? I thought you couldn't tell someone they couldn't have a service animal?

Ok, thank you ladies that helped and were kind about it. And I am not "trying to pass her off as a service animal", it's not "wrong" and you could have been much nicer about it. I thought since does perform assistance with my depression and anxiety greatly that she did qualify as a service animal. I didn't know the difference was so great. It was simply a question and I wasn't trying to pass her off as anything.