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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?

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It seems to me that elliptic curves came, as most things did, out of the study of elliptic integrals. –  Dylan Moreland Jun 10 '12 at 19:51
    
See livetoad.org/Courses/Documents/132d/Notes/… –  lhf Jun 10 '12 at 19:53
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Well, when you ask "when was the notion formalized," it really depends on which notion you mean (over the real numbers? The complex numbers? Over a field? An arbitrary base scheme?). Anyway, Dieudonne's The historical development of algebraic geometry should be good context for you, and I also like the discussion at the beginning of Stevenhagen's notes on elliptic curves (websites.math.leidenuniv.nl/algebra/ellcurves.pdf). –  Qiaochu Yuan Jun 10 '12 at 20:23
    
@lhf I saw that paper. It's a little too fuzzy for my taste. –  Eugene Jun 10 '12 at 20:27
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The one that is currently fairly well known ie genus one smooth nonsingular projective curve with a point $\mathcal{O}$. I guess that would be harder to find though so maybe whatever's relavant would be fine. –  Eugene Jun 10 '12 at 20:40

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I just finished taking a graduate course in Elliptic curves from Ezra Brown here at Virginia Tech. He loves to write expository articles in mathematics and I just realized (not surprisingly) that he's a co-author in the article cited by Alvaro. You might want to take a look at some of his other articles, here's another one: http://www.math.vt.edu/people/brown/doc/dioellip.pdf

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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.

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You may want to have a look at this one:

A. Rice and E. Brown, “Why Ellipses Are Not Elliptic Curves,” Mathematics Magazine 85 (2012), 163–176.

From their article:

We will therefore take a stroll through the history of mathematics, encountering first the ellipse, moving on to elliptic integrals, then to elliptic functions, jumping back to elliptic curves, and eventually making the connection between elliptic functions and elliptic curves. We will then finally be in a position to find out why no elliptically-shaped planar curves may ever be called elliptic curves.

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Why? What does it say? Does it suggest a history of elliptic curves? –  mixedmath Jun 14 '12 at 14:02
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From their article: "We will therefore take a stroll through the history of mathematics, encountering first the ellipse, moving on to elliptic integrals, then to elliptic functions, jumping back to elliptic curves, and eventually making the connection between elliptic functions and elliptic curves. We will then finally be in a position to find out why no elliptically-shaped planar curves may ever be called elliptic curves." –  Álvaro Lozano-Robledo Jun 14 '12 at 14:13
    
I say this because I think a little justification as to why someone should look for an article would greatly enhance your answer, not because I think I am the only person who would like to know. But with that, it seems to be a good find. –  mixedmath Jun 14 '12 at 14:17
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@mixedmath you are completely right, I should have added some sort of abstract... except that there is no official abstract to their article. However, that paragraph captures their goals. –  Álvaro Lozano-Robledo Jun 14 '12 at 14:24

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