In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying Diophantine equations. For instance it is possible to be "using" them implicitly and proving facts about them without actually knowing the formal concept like Ramanujan did with modular forms.
So my question is what is the history behind elliptic curves? When was the notion formalized and by whom? Any references to this?