= Cusp
{disambiguate=automorphic form}
{title2=$\Gamma\backslash\mathbb P^1(\mathbb Q)$}
For a finite-index subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbb Z)$, a <cusp (automorphic form)> is a $\Gamma$-equivalence class of rational boundary points, including infinity:
$$
\Gamma\backslash\mathbb P^1(\mathbb Q).
$$
Geometrically it describes a noncompact end of the quotient of the <upper half-plane>. Adding these classes compactifies the corresponding <modular curve>. The added points describe boundary data of the quotient.<Modular Functions and Modular Forms by James S. Milne>[section 2, cusps and the compactification of a finite-index quotient]{ref}
The full <modular group> has one cusp: a reduced rational number $a/c$ is the image of infinity under a matrix $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$ with determinant one. In contrast, $\Gamma_0(2)$ has inequivalent cusps represented by infinity and zero. A matrix sending infinity to zero must have $a=0$, hence $-bc=1$ and odd $c$, so it cannot belong to $\Gamma_0(2)$.
At infinity the <cusp width> is the least positive $h$ for which $z\mapsto z+h$ belongs to the effective stabilizer. The local coordinate $q=e^{2\pi iz/h}$ tends to zero as $y\to\infty$. Filling in $q=0$ adds one point. At other cusps one first changes coordinates by a matrix carrying infinity to the chosen rational boundary point. Growth and <constant term of an automorphic form> conditions must be checked at each inequivalent cusp.
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