It's my first question. I am studying elliptic function, Among them, I have a question about the Weierstrass elliptic function. In the Weierstrass elliptic function, their coefficients are expressed by 'Eisenstein series'. So I was so curious about origin of eisenstein series. When I search about it, 'Eisenstein series' was discovered by 'Kronecker-Weber theorem' and 'Jugendtraum'. But I want to know about this more detail.
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1$\begingroup$ "Jugendtraum" is a youthful dream, so I doubt this is relevant. I have tried to find the original publication(s) in which Gotthold Eisenstein introduced these series, so far without success. This is outside my area of mathematical expertise, and I suspect that Eisenstein's own notation may have differed significantly from the modern one, making it difficult to identify the original source (in German / French). $\endgroup$– njuffaCommented Apr 8, 2023 at 6:15
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3$\begingroup$ The relevant publication is likely: Gotthold Eisenstein, "Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenhängenden Doppelreihen (als eine neue Begründung der Theorie der elliptischen Functionen, mit besonderer Berücksichtigung ihrer Analogie zu den Kreisfunctionen)." Crelle’s Journal 35 (1847): 153-274. $\endgroup$– njuffaCommented Apr 8, 2023 at 6:27
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1$\begingroup$ Crelle's Journal is formally entitled Journal für die reine und angewandte Mathematik, and a scan of volume 35 from 1847 is available online at the Göttingen Digitization Center. The table of contents reveals that Eisenstein's publications (mostly on elliptic functions) took up the entirety of issues 2 and 3 (out of the four issues comprising volume 35), $\endgroup$– njuffaCommented Apr 8, 2023 at 6:42
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1$\begingroup$ Note that the Kronecker-Weber theorem dates to 1853, the year after Eisenstein's death in 1852. So it would seem completely irrelevant to the origin of Eisenstein's work on elliptic functions, which he appears to have created largely from scratch, based on a quick perusal of his publication. $\endgroup$– njuffaCommented Apr 8, 2023 at 7:06
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3$\begingroup$ Jugendtraum was Kronecker's conjecture that all Abelian extensions of imaginary quadratic field are generated by division values of suitable elliptic functions, described in his 1880 letter to Dedekind. It was a generalization of the Kronecker-Weber theorem and became Hilbert's 12th problem. Kronecker thought that Eisenstein's series would help prove it, see Elliptic Functions According to Eisenstein and Kronecker. Here is a link to Eisenstein's original 1847 paper, p.195:185ff. $\endgroup$– ConifoldCommented Apr 8, 2023 at 7:52
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2 Answers
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Andre Weil, Elliptic functions according to Eisenstein and Kronecher, Springer-Verlag, NY 1976.
answered Apr 8, 2023 at 12:13
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$\begingroup$ Thank you for your answer, I will search it $\endgroup$– pokssinCommented Apr 8, 2023 at 22:16
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See the book "Elliptic and Modular Functions from Gauss to Dedekind to Hecke" by Ranjan Roy. It is a very rich source of material on the history of topics related to modular functions. Using this book I was able to answer a question I once asked here about the origin of the Jacobi product formula for the discriminant modular function $\Delta(t)$: see here.
