Modular form

Remarks

Condition 2 with $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ gives $f(\tau + 1) = f(\tau)$, so $f(\tau) = \sum_{n \in \Z} a_n q^n$ with $q = e^{2 \pi i \tau}$. Condition 3 then says $a_n = 0$ for $n < 0$; this is "holomorphic at infinity". Example: every constant function is a modular form of weight $0$. Example: the Eisenstein series $G_4$ is a nonzero modular form of weight $4$. Non-example: the j-invariant satisfies 1 and 2 with $k = 0$, but $j = q^{-1} + 744 + \cdots$ is unbounded as $\operatorname{Im} \tau \to \infty$, so 3 fails. If $k$ is odd, the matrix $-I$ gives $f = (-1)^k f$, so $f = 0$. For a congruence subgroup $\Gamma$, condition 2 is required only for matrices in $\Gamma$. Condition 3 is then required at every cusp: $(c\tau + d)^{-k} f(\gamma \tau)$ is bounded on $\operatorname{Im} \tau \ge 1$ for every $\gamma \in \mathrm{SL}_2(\Z)$. The modular forms of weight $k$ form a finite-dimensional complex vector space $M_k$.