Doubt in deducing property of modular function

Question

While studying number theory from Apostol's Modular functions and Dirichlet Series in Number Theory, I am having problem in deducing 1 statement.

Statement is - An entire modular form of weight 0 is a modular function and since it is analytic everywhere including the point i $\infty $ , so it must be constant.

It is easy to deduce that a constant function of weight 0 is modular function and also by definition of entire modular forms it is analytic everywhere in H including point i$\infty $ . But why it must always be constant?

Does it omits any value?

Can someone please give a hint.

Answer 1

A modular form $f$ on $\mathrm{SL}(2, \mathbb{Z})$ that is holomorphic at $i\infty$ is bounded as $\mathrm{Im}(z) \to \infty$. Then it is clear that $f$ is bounded on the fundamental domain, as this is the union of a compact region and a neighborhood of $i\infty$. As $f$ is weight $0$, modularity implies that $f$ is bounded on the entire upper half-plane.

Since you are reading Apostol, you can now rely on the work he expressed in Chapter 2. In particular, a modular form of weight $0$ is what he calls a modular function, and Theorem 2.6 in the book is precisely what explains that $f$ is constant.

Answer 2

Cover the modular curve $$\bigcup_{\Im(\tau) > 0} SL_2(\Bbb{Z})\tau \cup SL_2(\Bbb{Z}) i\infty$$ by finitely many charts $\phi_j$ from closed disks (namely $\phi_1(z) = SL_2(\Bbb{Z})\frac{\log z}{2i\pi}$, $\phi_2(z) =SL_2(\Bbb{Z})( z+i)$ for $|z|\le 1-10^{-3}$)

If $f$ is $SL_2(\Bbb{Z})$ invariant, analytic on $\Im(\tau) > 0$ and bounded as $\tau \to i\infty$ then it attains its maximum modulus at some point $\phi_j(z_0)$ so that $f\circ \phi_j(z)$ is analytic and its modulus has a local maximum at $z_0$ thus it is constant and by analytic continuation so is $f$.