Where do the product expansions of modular forms come from?

Question

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?

Answer 1

Many "natural" examples of automorphic infinite products (also known as Borcherds products) can be explained using the singular theta lift of Harvey-Moore and Borcherds. These examples have the property that the exponents of the product are coefficients of a modular form.

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".