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Volume 3 of Gauss's werke contains an unpublished treatise with the title "Theory of new transcendents" (p.433-481 of the same volume), whose writing is dated, according to the editor Schering, to 1808; Schering states that those investigations prompted a communication to Schumacher. On page 441 of the same volume appears an interesting identity on theta functions which i suspect is a result of Gauss which i tried to locate in his writings for a very long time. The simplest way to describe it is to quote Gauss:

The division of seven leads to the following equation: $$b = (\frac{1-2x+2x^4-...}{1+2x+2x^4+...})^2$$ $$A = (\frac{1+2x^7+2x^{28}+...}{1+2x+2x^4+...})^2$$ $$B = (\frac{1-2x^7+2x^{28}-...}{1+2x+2x^4+...})^2$$ then: $$A^8-\frac{4}{7}A^6+\frac{16-32b^2}{49}A^5-\frac{30}{343}A^4+\frac{32-64b^2}{2401}A^3-\frac{2+768b^2-768b^4}{16807}A^2+\frac{48-2144b^2+6144b^4-4096b^6}{823543}A -\frac{1}{823543} = 0$$ If we replace $b^2$ with $1-b^2$ and $A$ with $-A$, the expression for $A$ is unchanged.

Gauss's fragment contains a similar result for $B$ but i didn't want to write it (it's very similar to the result for $A$). Note also that the denominators in all of the terms are powers of $7$.

Rewriting Gauss's indentities in modern notation: $$b = (\frac{\theta_4(0,x)}{\theta_3(0,x)})^2$$ $$A = (\frac{\theta_3(0,x^7)}{\theta_3(0,x)})^2$$ $$B = (\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$$

where $\theta_3(0,x)=1+2\sum_{n=1}^{\infty} x^{n^2},\theta_4(0,x) = 1+2\sum_{n=1}^{\infty}(-1)^n x^{n^2}$ are Jacobi's theta functions. Then Gauss found an inhomogenous algebraic (polynomial) expression $P(A,b) = 0$. This long presentation of Gauss's fragment was intended to present his results as accurate as possible. In a previous post (Question about Hermite's 1858 solution to the quintic equation using elliptic modular functions and it's relation to Gauss' and Jacobi's work), one of my questions was where Gauss's modular transformation of order 7 (according to some books, modular transformations of order 3,5 and 7 were known to Gauss since 1808). This fragment of Gauss deals with theta functions (not elliptic integrals), but since elliptic integrals and theta functions are related, i suspect this is the identity i searched for.

Therefore, my questions are:

  • Can anyone familiar with the theory of elliptic integrals (and theta functions) confirm that is the so-called modular transformation of order 7?
  • does this identity contain arithmetical information? i ask that because on another previous post (Did Gauss know Jacobi's four squares theorem?) i refered to another identity of Gauss (on p.445 in the same treatise) on an expansion of $\theta_3^4(0,x)$ in infinite series, and this identity does yield arithmetic information (namely Jacobi's four squares theorem).

And i apologize once again about my endless questioning about Gauss (i simply can't find references to many results in his writings...)!

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  • $\begingroup$ In order to aim those who want to help me, i think that in order to answer this question one needs, as a first step, to search for identities related to "lifting" of theta functions from $q$ to $q^n$, i. e to express $\theta(0,q^n)$ in terms of $\theta(0, q) $. I tried to search for a textbook on such "liftings" but i didn't find anything. $\endgroup$ – user2554 Aug 2 '20 at 21:47
  • $\begingroup$ The identities for "lifting" is very similar to my "Dedekind eta-function product identities". Let $\,t_n:=θ_3(0,q^n).\,$ Verify that $\,2\,t_4^2 − 2\,t_4t_1 + t_1^2 − t_2^2 = 3\,t_9^3t_1 − 3\,t_9^2t_1^2 + t_9t_1^33 − t_3^4 = 0.\,$ These are just examples of many such identities and not hard to find with a algebraic relation finding tool. $\endgroup$ – Somos Feb 15 at 2:11
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The formulas mentioned in Gauss's note are not the modular equation of order 7, but a very close one; the formula for $A$ is in fact the "Multiplier equation" .The point is that confirming that $P(B,b) = 0$ (not $P(A,b)$; the equation $P(B,b) = 0$ was not written by me in the posted question) is the modular equation of order 7 would be much easier, and the answer would be yes, if only Gauss would define $B$ to be $(\frac{\theta_4(0,x^7)}{\theta_3(0,x^7)})^2$ instead of $(\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$. To show the equivalency with the so-called "Transformation problem" of elliptic integrals, let us recall a few definitions and facts.

Definitions:

  • The nome $q$ of a theta function is denoted $x$ in Gauss fragment.
  • The modulus $k$ of a complete elliptic integral of the first kind is the constant parameter in $K(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1-k^2sin^2(\phi)}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-kt^2)}} $. The complementary modulus is defined to be $k' = \sqrt{1-k^2}$.
  • A complete elliptic integral of the first kind can be written as quotient of theta functions. The complementary elliptic modulus $k'$ can be written as: $(\frac{\theta_4(0,q)}{\theta_3(0,q)})^2$ where the nome $q$ is $e^{-\pi\frac{K(k')}{K(k)}}$.
  • The transformation problem, or the problem of modular equation determination, is for one modulus $k$, to find a second modulus $l$ such that $\frac{K(l')}{K(l)} = n\frac{K(k')}{K(k)}$ where $n$ is a positive integer.

The modular equation of order n:

Using the third fact, one can write $(\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2 = (\frac{\theta_4(0,e^{-n\pi\frac{K(k')}{K(k)}})}{\theta_3(0,e^{-n\pi\frac{K(k')}{K(k)}})})^2$. Rewriting $n\frac{K(k')}{K(k)}$ as $\frac{K(l')}{K(l)}$, one gets that $(\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2$ is nothing else than $l'$. Therefore an implicit relation between $k' = (\frac{\theta_4(0,q)}{\theta_3(0,q)})^2$ and $l' = (\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2$ is indeed the modular equation of order $n$: $P(k',l')=0$.

The connection to Gauss's fragment:

Unfortunately, as i remarked in the begining, Gauss defines $B$ to be $(\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$ so the whole argument doesn't correspond to Gauss's note. However, since he states the results for both $A$ and $B$, and since $\frac{B}{A} = (\frac{\theta_4(0,x^7)}{\theta_3(0,x^7)})^2$, one can say that at least numericaly, Gauss's results enable to determine $l'$ in terms of $k'$.

The relation of Gauss's fragment to the "multiplier equation":

According to p.512 volume 2 of the book series "Die elliptischen Funktionen und ihre Anwendungen" by Robert Fricke, the correct "Multiplikator gleichungen" (german for: "Multiplier equation") of order 7 is:

$$M^8 - 28M^6 + 112(1-2k^2)M^5 - 210M^4 + 224(1-2k^2)M^3 - (140+5376k^2k'^2)M^2 + (1-2k^2)(48-2048k^2k'^2)M - 7 = 0$$

Subsituting $M = 7A$ in the correct equation and dividing by $7^8 = 7\cdot 823543$, one gets Gauss's equation. So it's 99% certain that Gauss's formula is in fact the multiplier equation. The only thing i don't understand is why $M = 7A$ - according to all sources i found $M$ is supposed to be equal to $A$.

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