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Let $\Delta$ be the unique normalized cusp form of weight 12 and level $1$. Then Jacobi's wel-known formula states: $$\Delta(z) = q \prod_{n=1}(1-q^n)^{24},$$ where $q=e^{2 i \pi z}$.

For a graduate class I am teaching on modular forms, I am looking for a conceptual proof of this formula, probably using the elliptic interpretation of $\Delta(z)$ as the discriminant of the Weierstrass equation $\mathbb C / \langle 1, z \rangle$. I believe that such a proof exists because Serre, in "cours d'arithmétique", mentions it, with a reference to a prewar paper in German by Hurwitz (Gesamm. Abh, III, no 62), that I was not able to find in my library. Serre himself gives a proof based on the quasi-modularity of the false Eisenstein series $E_2$, and I known another proof from Apostol's textbook, attributed to Siegel, and that uses a clever computation of residues for the $\eta$ function. Both are interesting, but never can be qualified as conceptual by modern standards.

Do you know a proof of Jacobi's formula, or a reference (if possible modern and accessible) for such a proof ?

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# A conceptual proof of Jacobi's product formula for $\Delta$ ?

Let $\Delta$ be the unique normalized cusp form of weight 12 and level $1$. Then Jacobi's wel-known formula states: $$\Delta(z) = q \prod_{n=1}(1-q^n)^{24},$$ where $q=e^{2 i \pi z}$.

For a graduate class I am teaching on modular forms, I am looking for a conceptual proof of this formula, probably using the elliptic interpretation of $\Delta(z)$ as the discriminant of the Weierstrass equation $\mathbb C / \langle 1, z \rangle$. I believe that such a proof exists because Serre, in "cours d'arithmétique", mentions it, with a reference to a prewar paper in German by Hurwitz, that I was not able to find in my library. Serre himself gives a proof based on the quasi-modularity of the false Eisenstein series $E_2$, and I known another proof from Apostol's textbook, attributed to Siegel, and that uses a clever computation of residues for the $\eta$ function. Both are interesting, but never can be qualified as conceptual by modern standards.

Do you know a proof of Jacobi's formula, or a reference (if possible modern and accessible) for such a proof ?