# Where do the product expansions of modular forms come from?

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion

$$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$

for the discriminant cusp form of weight $12$ and level $1$. Another example is the cusp form of weight $2$ and level $11$

$$f(q) = q\prod_{n=1}^\infty(1-q^n)^2(1-q^{11n})^2,$$

which is attached to the elliptic curve $X_0(11) : y^2-y = x^3-x^2$. Such product expansions can be derived from the product expansion of the Dedekind $\eta$ function, by taking suitable combinations.

But why should such product expansions exist? Is there a reason to expect that they should exist, say, from the point of view of Galois representations?

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The Mellin transform provides a bridge between modular forms and Dirichlet series, which in turn can be expressed as Euler products. It may help to view the pair (Mellin transform, multiplicative convolution algebra) as analogous to the pair (Fourier transform, additive convolution algebra). –  Steve Huntsman May 3 '13 at 14:39
Dear @Steve: that's true, but I don't think the above products are directly related to the Euler products (they're products over $n$, rather than over $p$). Different animals! I may be wrong, though. Regards, –  Bruno Joyal May 3 '13 at 15:18
Similar question: mathoverflow.net/questions/108552/… There was no answer on this site, but you can find an answer on this blog: galoisrepresentations.wordpress.com/2012/10/26/… –  Joël May 3 '13 at 15:27
All power series with integer coefficients can be written formally as products of this type with integer exponents. –  Will Sawin May 3 '13 at 16:38
My understanding is that the connection is linked to lattice theory and discriminants of lattices. –  Daniel Parry May 4 '13 at 2:25

For example, the 24 in the exponents of the product formula for $\Delta$ can be matched with the positive-power coefficients of $12 \theta(0;\tau) = 12 + 24q^{1/2} + 24 q^2 + 24 q^{9/2} + \cdots$ in the following way: Shimura gave a correspondence between forms of weight $k + 1/2$ on $\Gamma_0(4)$ (satisfying some conditions) and forms of weight $2k$ for $SL_2(\mathbb{Z})$ (for some character), taking $f(\tau) = \sum_n c(n)q^n$ to $-c(0)B_k/2k + \sum_n q^n \sum_{d|n} d^{k-1} c(n^2/d^2)$. While this really works best for $k$ positive and even, you can remove the infinite constant term when $k=0$ to get something almost modular. Applying this to $12 \theta$, you naturally get $\log (\Delta/q)$.
In other words, 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. For example, the Koike-Norton-Zagier formula: $$j(\sigma) - j(\tau) = (p^{-1} - q^{-1}) \prod_{m,n>0} (1-p^m q^n)^{c(mn)}$$ arises as a lift of $j(\tau) - 744 = \sum_n c(n) q^n$ to $O(2,2)$, and the usual method of lifting involves a divergent integral of $(j-744)\theta$ over a fundamental domain of $SL_2(\mathbb{Z})$.
In general, I don't think these products are very naturally related to Galois representations. It is easy to lift forms like $-12\theta$ to get products like $1/\Delta$ of negative weight, which are somewhat invisible to the Langlands program (as far as I know). Instead, the products tend to show up naturally in subjects related to string theory, like the representation theory of infinite dimensional Lie algebras. For example, the product expansion of $1/\Delta$ gives the partition function of free bosons propagating in 24-dimensional space, and the Koike-Norton-Zagier formula is the Weyl denominator formula for the Monster Lie algebra. Borcherds has some expository overviews of this subject on his web page, e.g., number 28: "Automorphic forms and Lie algebras".