What does holomorphic at the cusp infinity means

Question

In the usual theory of classical modular forms, the modular forms defined to be "holomorphic at the cusp infinity". I do not know what this should mean? can anyone explain it for me?

Thanks

Answer 1

For simplicity, let's just consider modular forms (of weight $k$) of the modular group $SL(2,\mathbb{Z})$, i.e. holomorphic functions $f \colon \mathbb{H} \to \mathbb{C}$ satisfying two other conditions, the second being the "holomorphicity at $\infty$" in which we're interested.

The first condition is that $f$ has a nice symmetry under the action of $SL(2,\mathbb{Z})$; that is, $$f \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z \right) = (cz+d)^k f(z)$$ and this must hold for all matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2,\mathbb{Z})$. If we take the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ in the above equation, then we get that $f(z) = f(z+1)$ i.e. our modular form $f$ is periodic, hence has a Fourier series expansion. Write $$ f(z) = \sum_{n=0}^{\infty} a_n e^{2\pi inz}. $$ A common convention is to make the change of variable $q := e^{2\pi inz}$ and to view $f$ as a holomorphic function in $q$. The function $z \mapsto e^{2\pi inz}$ is a conformal map sending the upper half plane $\mathbb{H}$ to the punctured unit disc, where the point $i\infty$ is sent to the origin. Therefore, the modular form $f$ can be viewed as the holomorphic function $q \mapsto \sum_{n=0}^{\infty} a_n q^n$, from the punctured unit disc.

How does this help us understand "holomorphicity at $\infty$"? The second condition in the definition of a modular form, namely this "holomorphic at $\infty$" condition, says that $f(q)$ can be extended to a holomorphic function on the whole unit disc, i.e. it can be analytically continued through the puncture at the origin. By Riemann's theorem on removable singularities, this occurs when $f(q)$ is bounded is a neighbourhood of the origin; so you will often see the "holomorphic at $\infty$" condition written as "bounded at $\infty$" instead.

Two comments on this approach: one doesn't need to translate everything to the unit disc (as I did above) if we view $f$ as a function from a subset of the Riemann sphere. Second, another way to characterize this "holomorphic at $\infty$" condition is to say that $f$ defines a holomorphic function on the Riemann surface $SL(2,\mathbb{Z}) \backslash \mathbb{H}$ (this is the classical modular curve, which has a cusp at $i\infty$).

An excellent, but advanced, reference for this material are the first few chapters of Milne's course notes on modular forms. They are freely available here.