**My Motivation:** I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to follow, so I'd prefer answers in terms of explicit matrix representations, though a concrete descriptions of cylindrical sets or something might be helpful too.


**Background:** Let $\mathbf{A}\_2$ be the moduli space of dimension 2 principally polarized abelian varieties. If you prefer, just think of it as the Siegel upper half space of $2\times 2$ symmetric matrices $Z=X+iY$ with $Y$ positive definite, quotiented out by $\mathbf{SP}\_4(\mathbf{Z})$. If one compactifies $\mathbf{A}\_2$ according to the Satake compactification, there is a rank 1 boundary component and a rank 2 boundary component. 


**Main Question:** What is the preimage- in Siegel upper half space - of a simple system of fundamental neighborhoods for the rank 2 boundary component of $\mathbf{A}\_2$, in terms of an explicit description for the matrix co-ordinates.


**Form of Answer that I'd like:** Something about $Y$ as a quadratic form, involving its determinant, minimal vector length, etc...


I'd strongly prefer an answer entirely specific to this case, if possible, and not something for general locally symmetric spaces. 


**Additional question I'd be grateful to get the answer to:** Same question but for the rank 1 boundary component.


Thanks!!



An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$
This answer is incomplete but I'll try to finish it later.


First note that you can't express the answer only using $Y$, even in the $g=1$ case. If the imaginary part of $\tau$ goes to $0$, you do not know anytihng about what $j(\tau)$ is doing.


I think you want to work with the induced quadratic form on pairs $(v,w) \in \mathbb Z^2 \times \mathbb Z^2 = \mathbb Z^4$ given by 


$$ (v\_1,w\_1) \cdot (v\_2,w\_2) = \operatorname{Re}\left( \overline{(v\_1 + Z w\_1)}^T Y^{-1}(v\_2 + Z w\_2) \right)$$


If I calculated correctly, this is the actual Riemann form of your torus, because it comes from a Hermitian form $Y^{-1}$ on $\mathbb C^2$ whose imaginary part resticts to the standard symplectic from on $\mathbb Z^4$.


The action of $SP\_{4}(\mathbb Z)$ on this is the natural action on quadratic forms in four variables. Indeed, $v^T Y^{-1} v$ on $\mathbb C^2$ is the unique Hermitian form whose imaginary part restricts to the standard symplectic form on the lattice $\mathbb Z^4$ embedded by the map $(v\_1, v\_2) \mapsto v\_1 + Z v\_2$. The action of $SP\_{4}$ on $Z$ comes from the action of $SP\_4$ on this lattice plus some weird action on $\mathbb C^2$, but since the action preserves the standard symplectic form it will preserve the uniquely determined Hermitian form.


Now I think a basis of neighborhoods of the rank $k$ component of the boundary of this lattice consists of points where there are $k$ vectors $v\_1,\dots,v\_k$ in $\mathbb Z^4$ that are linearly independent, whose pairing under the symplectic forms with each other all vanish, and whose lengths under the quadratic form are all $<\epsilon$. 


I think this because if you have a sequence of such quadratic forms with vectors $v\_1,\dots,v\_k$ so that the length of those vectors goes to $0$, you obtain a lattice of rank $4-2k$ with a symplectic form by modding out by $v\_1,\dots,v\_k$ and taking the kernel of the map to $v\_1,\dots, v\_k$, and you obtain a corresponding quadratic form by taking the limit of the original quadratic form, which will be well-defined as the lengths of $v\_1,\dots,v\_k$ goes to $0$. So a sequence of points that are in all the rank $k$ neighborhoods but not all the rank $k+1$ neighborhoods will have an accumulation point in $\mathcal A^{2-k}$. That is precisely what the Satake compactification is supposed to look like. I will try to rigorously justify this...


In fact the condition that $v\_1,\dots,v\_k$ have zero pairing under the symplectic form is irrelevant. As the symplectic form and quadratic form come from the same Hermitian forrm, they are related by a Cauchy-Schwartz inequality, so if $v\_1,v\_2$ have length at most $\epsilon$, their pairing is at most $\epsilon^2$, so because the symplectic form is integral it is $0$. So we can express the condition only in terms of tuples of short vectors in the quadratic form with matrix


$$\begin{pmatrix} Y^{-1} & Y^{-1} X \\ X Y^{-1} & XY^{-1}X + Y\end{pmatrix}$$


That's totally explicit but might not be too fun. An alternate desciption of the system of neighborhoods of the rank $2$ part which may or may not be easier is the $SP\_4(\mathbb Z)$ orbit of the set of pairs $X,Y$ where the shortest vector of $Y$ has length at least $1/\epsilon$. This is the same because you can always transform your two short vectors to be the first two basis vectors.


