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Gauss's results on the interconnection between the different values of the arithmetic-geometric mean of two complex numbers as recorded in his private notebooks led him to introduce foundational aspects of the theory of modular functions, and is perhaps the deepest part of his work in this area (as far as "modular forms" could be concieved in the early 1800s). This work is the only trace of such mathematical activity for about 4-5 decades (Abel and Jacobi were more concerned in the algebraic developement of the theory and not in the basic conception of modular forms), and is therefore very interesting historically.

In connection with this work of Gauss, I found an English translation of some of Gauss's fragments - just google "Posthumous Fragments on the Theory of the Arithmetic-Geometric Mean and the Modulus Function" and it will appear as one of the first search results. These translated fragments appear to be very deep, especially the parts on the "Construction of the continuous fraction" and "The Quadratic Form. The Fundamental Domain". In addition, the translation refers to the exact pages in Gauss's werke from which they were translated.

However, going into the same pages which the english translation refers to reveals that those fragments are not there, and in fact are not extant at all in Gauss's collected works (for example, the figure on p.25 of the translation cannot be found)! Therefore, i'd like to know what is the reference which those translated fragments are based on.

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    $\begingroup$ Perhaps there is a better chance of getting a response on Math Overflow. $\endgroup$ – Conifold Feb 1 at 18:58
  • $\begingroup$ @Conifold - generally i dont like to ask questions which are purely historical on Mathoverflow, but i think you are right in this case (maybe i'll somehow get a response from the translators of those fragments...). I'll do this soon. $\endgroup$ – user2554 Feb 1 at 22:21

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