 Cusp (automorphic form)

For a finite-index subgroup $\Gamma$ of ${\mathrm{SL}}_2(\mathbb{Z})$, a cusp is a $\Gamma$-equivalence class of rational boundary points, including infinity: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.^[ref]

The full modular group has one cusp: a reduced rational number $a/c$ is the image of infinity under a matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ 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.