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Elliptic integrals, elliptic functions, and elliptic curves are well-studied objects in mathematics. Unfortunately, for various reasons, they are usually not covered in undergraduate courses. I feel that elliptic integrals and elliptic functions have been neglected in the current academic trends, although I might be mistaken in this regard, except perhaps in certain specialized subfields.

  • Are there any textbooks written in modern language for undergraduates and beginning graduate students that cover these topics?

Especially, I would like to see their historical development, classifications, relationships between them, and modern perspective.

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  • $\begingroup$ Did you do a web search? By "modern language" do you mean English? What do you mean by "modern perspective"? $\endgroup$ Commented Dec 22, 2023 at 18:13
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    $\begingroup$ This AMS article by Burt Totaro could be a start. I liked very much reading it a bit and it has 45 references. $\endgroup$ Commented Dec 22, 2023 at 18:31
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    $\begingroup$ Not a textbook, but a very readable introduction to elliptic curves and the Birch and Swinnerton-Dyer conjecture: Elliptic Tales: Curves, Counting, and Number Theory, by Avner Ash and Robert Gross. $\endgroup$ Commented Dec 22, 2023 at 23:34
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    $\begingroup$ Some of the references in my answer to How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$? might be helpful. FYI, I know very little about elliptic functions (and this only from their appearance in some upper level and beginning graduate level physics courses I took during the late 1970s and early 1980s), but I've always wanted to pursue the very beginnings of the classical study of elliptic integrals, and that answer was the result of 2 weeks of my first nontrivial attempt -- I happened to have 2 weeks mostly free and the desire. $\endgroup$ Commented Dec 23, 2023 at 6:53
  • $\begingroup$ I would recommend my own blog series on this topic. It covers elliptic integrals, Jacobian elliptic functions, Jacobian theta functions, and Weierstrass elliptic functions. $\endgroup$ Commented Dec 28, 2023 at 15:53

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How about H. McKean and V. Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press (1999) publisher's page

Blurb:

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows a historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics.

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