TITLE: History of elliptic curves
QUESTION [20 upvotes]: 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?

REPLY [15 votes]: Clebsch, in the 1860s, proved that curves of genus 0 are parametrized by rational functions, and that those of genus 1 are parametrized by elliptic functions. Juel gave a geometric
interpretation of the group law in the 1890s, Poincare asked in 1901 whether the rational points on a curve of genus 1 are finitely generated, and Mordell proved this in the 1920s. 
As for examples, integral solutions of $y^2 = x^3 - 2$ etc. were determined (without proof)
by Fermat, and Euler later solved it using algebraic numbers. There's a whole industry of
mathematicians who tried so solve such equations at the end of the 19th century (Lucas, Sylvester, B. Levi, etc.).  
The modern theory took off in the 1930s with Hasse's work on the number of points on elliptic curves over finite fields, which subsequently was generalized by Weil with his conjectures.