A conceptual proof of Jacobi's product formula for $\Delta$ ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:08:18Z http://mathoverflow.net/feeds/question/108552 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108552/a-conceptual-proof-of-jacobis-product-formula-for-delta A conceptual proof of Jacobi's product formula for $\Delta$ ? Joël 2012-10-01T15:43:14Z 2012-10-01T16:18:36Z <p>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 $q=e^{2 i \pi z}$.</p> <p>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.</p> <blockquote> <p>Do you know a proof of Jacobi's formula, or a reference (if possible modern and accessible) for such a proof ?</p> </blockquote>