Did Jacobi prove the product formula $\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$ for the modular discriminant function, and if so where?
I have tried without success to track down references (other than looking through Jacobi's collected works), but when I search for terms like "Jacobi product" I get lots of hits on his triple product formula, and inserting "discriminant" into the search is not helping on isolating a reference that would discuss Jacobi's own work on this.
I found an answer to my own question from Ranjan Roy's book "Elliptic and Modular Functions from Gauss to Dedekind to Hecke".
On page 293, Roy says the $q$-product of the $\Delta$-function is not due directly to Jacobi, but to Klein (and ealier to Weierstrass: see below). In Klein's 1878/1879 paper Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades (pp. 13-75 in Volume 3 of his collected works here), which has some really beautiful pictures, $\Delta$ is defined on the top of the 3rd page as a cubic discriminant and in equation (18) on the ninth page you can see the $q$-product for the 12th root of $\Delta(\omega)$, where Klein's $q$ is $e^{\pi i\omega}$ instead of $e^{2\pi i\omega}$. Taking $12$th powers of both sides of the equation yields the usual product for $\Delta$. I think the notation $\Delta$ for the discriminant function is due to Klein.
The second part of Klein's equation (17) expresses the $12$th root of $\Delta$ in terms of a product of three functions whose cube he expresses just above his equation (18) as a $q$-product that is taken from equation (2) on p. 89 of Jacobi's Fundamenta Nova here. That's the reason for attributing the $q$-product of the $\Delta$-function to Jacobi. Since Klein's work relied on Jacobi's work, Klein needed knowledge of elliptic functions.
A year before Klein, in 1877, Dedekind introduced the related function $\eta(\omega)$ here: a definition and its transformation under ${\rm SL}_2(\mathbf Z)$ are equations (3) and (6) on p. 281 and its product representation $q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$ is equation (24) on p. 285, with $q = e^{\pi i\omega}$. Dedekind's paper does not mention the discriminant function $\Delta$. It also has no pictures.
Roy says earlier in his book, on p. 99, that Weierstrass used his sigma function to derive the $q$-product of $\Delta$ in his lectures on elliptic functions in 1862-1863: see equation 12 on p. 164 of the 5th volume of Weierstrass collected works here, where $G$ is $\Delta$ up to powers of $2$.
Hurwitz, who was a PhD student of Klein, was the first to develop modular functions without needing prior knowledge of elliptic functions (or theta functions) of any kind in 1881, and in particular he proved the product for $\Delta$ without needing elliptic functions; see p. 552 here. This is the approach to modular functions in introductory book today, using Eisenstein series as a starting point.
