automorphic form

《A First Course in Modular Forms》 · 第 72 页 · Section 3.2「Automorphic forms」，Definition 3.2.1

Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb Z)$ and let $k$ be an integer. A function $f:\mathcal H\to\mathbb C$ is an automorphic form of weight $k$ with respect to $\Gamma$ if (1) $f$ is meromorphic, (2) $f$ is weight-$k$ invariant under $\Gamma$, and (3) $f[\alpha]_k$ is meromorphic at $\infty$ for all $\alpha\in SL_2(\mathbb Z)$.