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Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique normalized cusp form of weight $12$ and level $1$), but unlike the previous question, I am asking for a proof specifically from the point of the view of the Leech lattice. For reference, that post contained the following 3 leads:

  • a reference to a "prewar paper in German by Hurwitz (Gesamm. Abh, III, no 62)" that no one seems to have found/analyzed/discussed yet;
  • Weil's "Sur une formule classique" (Collected works vol III pp 198-200) using a version of the "converse theorems";
  • an argument from Frank Calegari (?) using "Hecke eigenvalues of the $E_2$ series": https://www.galoisrepresentations.com/2012/10/26/jacobi-by-pure-thought/

A similar MO thread Where do the product expansions of modular forms come from? mentions also

  • "products like $\Delta$ arise by exponentiation of a Howe theta lift (of which Shimura's correspondence is a special case), although the lift may need to be regularized."

One comment in the latter MO thread mentions connections to lattices. Remark 40 in these notes of Tao also mentions that the product formula for $\Delta$ (now referring to this normalization) is an "interpretation (after some additional effort)" of the relation $E_{12} - \frac{65520}{691}\Delta = \Theta_\Lambda$ where $\Theta_\Lambda(\tau):= \sum_{x\in \Lambda} e^{-i\pi \tau \|x\|^2}$ is the theta-function of the Leech lattice $\Lambda \subseteq \mathbb R^{24}$.

When I tried Googling this connection, I only came across https://math.stackexchange.com/questions/4493108/what-is-precisely-the-connection-between-the-leech-lattice-and-the-dedekind-eta, which links some slides of John Baez, though I was not very satisfied with these answers (the post https://math.ucr.edu/home/baez/week125.html linked in the Credits provided some further insight).

Question: what exactly is the "some additional effort" needed to go between the relations $\Delta = (2\pi )^{12} \eta^{24}$ and $E_{12} - \frac{65520}{691}\Delta = \Theta_\Lambda$ alluded to above in Tao's Remark 40? I would really prefer an explanation which doesn't go into string theory/monstrous moonshine territory, if at all possible.

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    $\begingroup$ Given that the space of modular forms of weight $12$ is linearly generated by $E_4^3$ and $E_6^2$, and that $\Delta$, $E_{12}$, and $\Theta_\Lambda$ are in that space, there must be a linear relation between the three of them. Two Fourier coefficients are enough to determine the linear relation. $\endgroup$
    – Somos
    Commented Dec 24, 2022 at 15:10
  • $\begingroup$ @Somos I'm aware. My question is about how to get from that relation to the eta function. $\endgroup$
    – D.R.
    Commented Dec 24, 2022 at 18:02

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Steve Carnahan in his answer to this question Where do the product expansions of modular forms come from? gives a conceptual explanation of the product form for $\Delta$, but it has nothing to do with the Leech lattice.

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  • $\begingroup$ I am already aware of Steve Carnahan's answer, and in fact quoted it in the text of my post. Like you yourself said, your answer has nothing to do with the question I asked (in the title, and in the bold at the bottom of my post). $\endgroup$
    – D.R.
    Commented Dec 25, 2022 at 19:35
  • $\begingroup$ You asked for a conceptual explanation of the product formula and Carnahan provided one. Now you seem to insist on a conceptual explanation involving the Leech lattice but I see no reason why there should be one. You quoted Carnahan, but it wasn't clear to me you understood his explanation. $\endgroup$
    – Jeff Harvey
    Commented Dec 25, 2022 at 23:48
  • $\begingroup$ Very well, I will clarify my question to make it unambiguous that I am looking for an explanation involving the Leech lattice. It is discouraging to hear that you "see no reason why there should be one"; I assumed Tao got his remark from a widely-known source, but perhaps I will have to ask him directly what he meant. $\endgroup$
    – D.R.
    Commented Dec 26, 2022 at 6:50
  • $\begingroup$ Scott Carnahan $\endgroup$
    – user108998
    Commented Dec 26, 2022 at 15:58

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