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I would like to know more about the history of the widely used terms "Weierstrass equation" and "Weierstrass normal form", as they appear in the theory of elliptic curves. When were these terms first coined? What did Weierstrass exactly prove? Was he first to prove that every elliptic curve can be written in this form? What papers or books of Weierstrass are relevant here?

A model for an elliptic curve $E$ of the form $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ is said to be a Weierstrass equation for $E$. A model of the form $y^2=f(x)$, where $f(x)$ is a monic cubic polynomial, is a model in Weierstrass normal form. (Classically, a Weierstrass normal form may refer to an equation of the form $y^2=4x^3-g_4x-g_6$, or one of the form $y^2=x^3+Ax+B$.)


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I always supposed his name was attached to this not because he used elliptic curves, but because he created the basic elliptic function $\wp(z)$ associated to any lattice and found the differential equation $\wp(z)$ and $\wp'(z)$ satisfy, which is a special case of the later-named Weierstrass equation. Possible references to look at are "Mathematics of the 19th Century Volume II" (starting at p. 220), Bottazzini's "The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass". –  KCd Apr 14 '12 at 17:58
Or adding to KCd's comment: the fact that the Weierstrass elliptic function satisfies that neat differential equation allows for its use in representing elliptic curves parametrically. (Also, I always thought it was a natural thing to "depress" cubics (i.e. remove the quadratic term through a substitution) before doing anything else with them.) –  Guess who it is. Apr 15 '12 at 15:33

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Perhaps this link can provide a starting point http://math.nist.gov/opsf/personal/weierstrass.html .

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Thank you for the link. There is also a nice biography here www-history.mcs.st-and.ac.uk/Biographies/Weierstrass.html but as in your link, they discuss his work on elliptic functions, and the concrete topic of weierstrass equations for elliptic curves is not mentioned (that I can see!). Certainly, elliptic functions are very much related to this question, but not exactly what I am looking for. –  Álvaro Lozano-Robledo Apr 14 '12 at 16:38

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