I'll begin by defining the notion of a cusp of a congruence subgroup of $\textrm{SL}_2(\mathbb{Z})$:
$\textrm{SL}_2(\mathbb{Z})$ has a natural action on the compactified upper half complex plane $\mathbb{H} := \{ z \in \mathbb{C} \mid \textrm{Im}(z)>0 \} \cup \{ \infty \}$, given by:
$$
g \cdot z := \frac{az+b}{cz+d}, \quad \textrm{for all}\quad g=\left(\begin{array}[cc]\ a & b\\ c & d \end{array}\right) \in \textrm{SL}_2(\mathbb{Z}),\quad z \in \mathbb{H}.
$$
Naturally, any subgroup of $\textrm{SL}_2(\mathbb{Z})$ inherits this action on $\mathbb{H}$.
We define the boundary of the compactified upper half plane as:
$$\partial (\mathbb{H}) = \{z \in \mathbb{C} \mid \textrm{Im}(z) = 0 \} \cup \{ \infty \} \simeq \mathbb{R} \cup \{ \infty \} \simeq \mathbb{P}^1(\mathbb{R}).$$
Now, let $\Gamma < \textrm{SL}_2(\mathbb{Z})$ be a Fuchsian group of the first kind (e.g. a congruence subgroup). A point $s \in \partial(\mathbb{H})$ is a cusp of $\Gamma$, if there exists a parabolic element $\gamma \in \Gamma$ such that $\gamma \cdot s = s$. Cuspidal elliptic modular forms of level $\Gamma$ are hence defined to be elliptic modular forms of level $\Gamma$ that vanish at the cusps of $\Gamma$.
Now, let $\Gamma$ again be some Fuchsian group of the first kind, and let $s \in \mathbb{P}^1(\mathbb{Q})$ be some cusp of $\Gamma$. As $\textrm{SL}_2(\mathbb{Z})$ acts transitively on $\mathbb{P}^1(\mathbb{Q})$, $\exists \sigma \in \textrm{SL}_2(\mathbb{Z})$ such that $\sigma(\infty) = s$. Now, let $\Gamma_s$ be the stabiliser of $s$. As the elements in $\sigma^{-1} \Gamma_s \sigma$ all fix $\infty$, they are all given by translations. The width $h_s$ of the cusp $s$ is then defined to be the smallest positive such translation. – It is easy to see that the notion of the width of a cusp is well-defined, as it is independent of the choice of $\sigma$.
I have some questions about Siegel's generalisation of the above. I will begin by defining my terms:
There exists a natural generalisation of the upper half complex plane, given by:
$$\mathbb{H}_n := \{ Z \in \textrm{Sym}_n(\mathbb{C}) \mid \textrm{Im}(Z)\ \textrm{is positive-definite} \},$$
where $n$ is some natural number and $\textrm{Sym}_n(\mathbb{C})$ denotes the vector space of symmetric $n \times n$ matrices over $\mathbb{C}$. We define the symplectic group of degree $d$ over $\mathbb{Z}$ as:
$$
\textrm{Sp}_{2d}(\mathbb{Z}) := \left\{ g \in \textrm{GL}_{2d}(\mathbb{Z}) \mid g^tJ_{2d}g = J_{2d} \right\},
$$
where $J_{2d} := \left(\begin{array}[cc]\ 0 & \textrm{Id}_d\\ -\textrm{Id}_d & 0 \end{array}\right)$, $\textrm{Id}_d$ denotes the $d \times d$ identity matrix, and $g^t$ denotes the transpose of $g$.
As before, we can define e.g. congruence subgroups of $\textrm{Sp}_{2d}(\mathbb{Z})$ given by:
$$
\Gamma^{(d)}(N) := \left\{ g \in \textrm{Sp}_{2d}(\mathbb{Z}) \mid g \equiv \textrm{Id}_{2d}\ (\textrm{mod}\ N) \right\}
$$
for some $N \in \mathbb{N}$.
I suppose a natural way of defining boundary of $\mathbb{H}_n$ is:
$$
\partial(\mathbb{H}_n) = \left\{Z \in \textrm{Sym}_n(\mathbb{C}) \mid \textrm{Im}(Z)=0 \right\}.
$$
And hence to define the cusps of a congruence subgroup of $\textrm{Sp}_{2d}(\mathbb{Z})$ to be the points in $\mathbb{H}_d$ that are fixed by an element in some class of non-scalar elements in the subgroup.
My question is, what is, in fact, the appropriate way of defining a cusp of a congruence subgroup of $\textrm{Sp}_{2d}(\mathbb{Z})$? And does there exist an analogy to the notion of the width of a cusp, as in the GL2 case? Almost all resources that deal with Siegel modular forms restrict themselves to Siegel modular forms of full level (i.e. defined over the full symplectic group). The only resource I have found that treats Siegel modular forms over arbitrary congruence subgroups is E. Freitag's book Singular Modular Forms and Theta Relations, but he defines a cuspidal Siegel modular form implicitly in terms of its Fourier expansion. Is anyone aware of the existence of other relevant resources?
Best Answer
The standard, classical way to define a cusp form for Siegel modular forms, is in terms of the Fourier expansion. If you want to define the cusps of a quotient $Y = \Gamma \backslash H^n$ of Siegel upper half space, you need to understand the "infinities" of this quotient.
When $n=1$, you basically get a Riemann surface with holes (holes = infinities), so the cusps are just a finite set of points. When $n > 1$, this is no longer the case. You have whole (lower-dimensional) "varieties" of ways in which you can approach infinity. Put another way, to compactify (in a nice way) you need to add more than just a finite set of points. (Note: the way you defined the boundary is not right, as you want Im $Z$ to not be positive definite, not 0.)
So basically the analogue of cusps will be $X - Y$, where $X$ is a nice compactification of $Y$. For more details, see, e.g., this survey:
The point is, it's complicated in higher dimensions, so if you're just working with Siegel modular forms, it's easiest to just formulate the notion of cuspidality in terms of Fourier expansions.