I'm studying the theory of elliptic curves and in all the books I've read they use the term "Weierstrass equation" or a similar one. But so far I've failed to find out when that term was used for the first time. In which Weierstrass' paper is included the general equation of an elliptic curve? Who was the first author to use that term to link the equation of an elliptic curve to Weierstrass?
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$\begingroup$ The same exact question was asked here: math.stackexchange.com/questions/131763/… $\endgroup$– KConradCommented Jan 23, 2013 at 22:06
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4$\begingroup$ On page 36 of Tate's article in Antwerp IV (see modular.math.washington.edu/Tables/antwerp/tate), he calls these equations that allow xy and y terms "generalized Weierstrass form". Since he uses the label "generalized", perhaps that means the terminology had not yet settled down. My hunch is that Weierstrass himself only worked with the equation $y^2 = 4x^3 - g_2x - g_3$, since that is satisfied by his $\wp$-function and its derivative, which is how he dealt with elliptic curves. Without arithmetic issues in mind, why look at something messier than $y^2 =$ cubic for ell. curves? $\endgroup$– KConradCommented Jan 23, 2013 at 22:28
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This must come from the name Weierstrass normal form given to the elliptic integrals $$ \int\frac{ds}{\sqrt{4s^3-g_2s-g_3}},\quad \int\frac{s\,ds}{\sqrt{4s^3-g_2s-g_3}},\quad \int\frac{ds}{(s-\alpha)\sqrt{4s^3-g_2s-g_3}} $$ in e.g. Klein (1885, pp. 454–459), Enneper-Müller (1890, pp. 26–30, 222), Burkhardt (1899, p. 161), Hensel-Landsberg (1902, p. 650), Kohn-Loria (1909, p. 480), Fricke (1913, pp. 253, 294, 297), etc.
(First publication of the normal form itself was by Weierstrass’ students: Biermann (1865, pp. 5–10), Müller (1867, pp. 1, 19), Schwering (1869, p. 9), Kiepert (1870, p. 7), Schwarz (1871, pp. 78, 102), Weierstrass-Schwarz (1885, pp. 2, 12, 31, 61, 68, 86).)
answered Aug 4, 2019 at 16:20
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$\begingroup$ The links go to reviews rather than the original papers, and none of the reviews use the term "Weierstrass normal form" (in German). $\endgroup$– KConradCommented Aug 4, 2019 at 16:42
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$\begingroup$ @KConrad The first two zbmath reviews have links to full text, at the bottom. $\endgroup$ Commented Aug 4, 2019 at 16:49
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1$\begingroup$ I see. Direct links to the first two papers at archive.org are archive.org/details/formelnundlehrs01weieuoft/page/n6 and archive.org/details/elliptischefunk00burkgoog/page/n11 $\endgroup$– KConradCommented Aug 4, 2019 at 17:36