Why is the j-invariant a modular function but not a modular form?

Question

I read that the j-invariant is a modular function but not a modular form. This is confusing, because the j-invariant doesn't have poles in the upper half plane. What is the difference between modular functions and modular forms anyway?

Answer 1

An important defining constraint for modular forms is boundedness as ${\rm Im}(\tau) \to \infty$. To see that this is not true of the $j$-function, from its $q$-expansion $$ j(q) = \frac{1}{q} + 744 + 196884q + 21493760q^2 + \cdots $$ where $q = e^{2\pi i\tau}$ for $\tau = x + iy$ with $y > 0$, so $|q| = e^{-2\pi y}$ and $|1/q| = e^{2\pi y}$, we have $|j(q)| \to \infty$ as ${\rm Im}(\tau) \to \infty$.

Because the $j$-function is an ${\rm SL}_2(\mathbf Z)$-invariant function, it resembles a modular form of weight $0$ if you ignore the boundedness condition for modular forms. And it certainly is not a modular form of weight $0$ because the modular forms of weight $0$ are constant functions.