You can't do that and they could come back at any time or they could tell the police that you stole their child and you would go to jail. If you want the child, you will have to get a lawyer and legally adopt the child (go through the adoption process which may include court). Don't do it illegally as you may end up losing the child and in jail.
Oh and don't put your name on the birth certificate. That would be falsifying a legal document (they could use dna to determine the real parents and they would know you were not).
So to answer your questions: you must go to court to adopt the child legally and you cannot put your name on the birth certificate until the court orders it (ie. after the adoption).
Good luck!
