How does Siegel's Hilbert-Blumenthal fundamental domain differ from Götsky's?

Question

This question has been changed to something related but different from the original question. Thanks to @paulgarrett for chatting with me and helping me hone in on a more interesting part.

The first steps toward constructing a fundamental domain for a Hilbert-Blumenthal (aka Hilbert modular) surface were published by Blumenthal (1901) and a way of completing the idea was published by Götsky (1927). Then Siegel (1943) introduced a different approach using an alternative metric that measures distance from cusps. I want to know exactly how Siegel's variation affects the resulting fundamental domain, as compared to the previous method. By way of motivation, Cohn (e.g. 1965) wrote a series of articles where he appeals to the original formulation because he found it to give a more intuitive sense of certain geometric properties, and I have a project where we improve on what Cohn was doing.

A Hilbert-Blumenthal (aka Hilbert modular) surface is defined as follows. Let $n\in\mathbb{N}$ be squarefree, let $K=\mathbb{Q}(\sqrt n)$, let $\mathbb{Z}_K$ be the ring of integers of $K$ and let $\Gamma_K:=\mathrm{PSL}_2(\mathbb{Z}_K)$. Let $\sigma:\Gamma_K\rightarrow\Gamma_K$ be the map that does $a+b\sqrt n\mapsto a-b\sqrt n$, $a,b\in\mathbb{Q}$ to the entries of a matrix. Let $\mathcal{H}^2$ be the upper half-plane model for the hyperbolic plane and for $\gamma\in\Gamma_K$ and $p\in\mathcal{H}^2$, let $\gamma(p)$ denote the action by Möbius transformation. Then $\Gamma_K$ acts discretely on the product $\mathcal{H}^2\times\mathcal{H}^2$ via $\gamma(p_1,p_2)=\big(\gamma(p_1),\sigma(\gamma)(p_2)\big)$. The quotient $M_K:=(\mathcal{H}^2\times\mathcal{H}^2)/\Gamma_K$ is called a Hilbert-Blumenthal surface. Some properties of $M_K$: it is an orbifold with quotient singularities and cusp singularities, it has finite volume, it has a finite amount of cusps equal to the class number of $K$, each cusp admits a neighborhood having the structure of $\mathbb{R}^+\times($a Sol $3$-manifold$)$ and all Sol $3$-manifolds occur in this way up to commensurability.

A fundamental domain for $M_K$ is a set of points in $\mathcal{H}^2\times\mathcal{H}^2$ that contains precisely one representative of each orbit of the action of $\Gamma_K$. For simplicity, let's focus on the case where $n\not\equiv_41$ and $K$ has class number $1$. If you feel you have an answer to my question for that case, you may not need the details below, but I'm including them for everyone else.

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The first classical approach to this is to think about the generators for $\Gamma_K$ $$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & \sqrt n\\ 0 & 1 \end{pmatrix}, \begin{pmatrix} \varepsilon & 0\\ 0 & \varepsilon^{-1} \end{pmatrix}, \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix},$$ where $\varepsilon$ is the fundamental unit of $\mathbb{Z}_K$. Denote a point in $\mathcal{H}^2\times\mathcal{H}^2$ by $(p_1,p_2)$. The group generated by the first two of these matrices has a fundamental domain $R$ like a rectangle cross $\mathbb{R}^+i\times\mathbb{R}^+i$, defined by $$-1\leq\Re(p_1)+\Re(p_2)<1\\ -\sqrt n\leq\Re(p_1)-\Re(p_2)<\sqrt n.$$ The group generated by the third matrix has a fundamental domain $W$ like a wedge cross $\mathbb{R}\times\mathbb{R}$ defined by $$\varepsilon^{-1}\leq\frac{\Im(p_1)}{\Im(p_2)}<\varepsilon.$$ And the group generated by the fourth matrix has a fundamental domain $C$ like the exterior of varying pairs of half-cirles (one in each $\mathcal{H}^2$), defined by $|p_1p_2|\geq 1$. What happens then is that the region $R\cap W\cap C$ contains a fundamental domain for $\Gamma_K$, and is equal to the fundamental domain only when $n=5$. When $n\neq5$, this region has infinite volume and so requires a lot of refinement.

Götsky's improvement on this is to replace $R$ with a set of equivalence class representatives that minimize the value $|p_1p_2|$. He shows that this results in a fundamental domain always, but one that is pretty difficult to describe, especially its boundary. Cohn's articles on the topic give a way of formalizing this, defining what it means for the boundary to be "simple," but even with that, the boundary is only simple when (again) $n=5$. (My current project improves on this.)

As for what Siegel does, I'm afraid I'm not capable of summarizing it. When I look at his original literature, it is excellently written, but very hard to pull pieces out of without reading the entire thing. On top of that, I do not have a strong background in number theory, just some of its applications to topology. Somewhat more accessible is chapter 3 of this book by van der Geer, but I fall out exactly in the 3rd paragraph of page 10 where he says "It is easy to see that...." If you check the surrounding text, he seems to be saying that with Siegel's metric, in the one-cusped case, a fundamental group for the cusp stabilizer is equal to a fundamental group for $\Gamma_K$, which makes absolutely no sense to me.

Answer 1

I'm not familiar with the Götsky--Cohn construction but Siegel's (as explained in van der Geer's book) seems clear:

• there is a "height function" $y$ on $X = \mathbb H^2 \times \mathbb H^2$ (the "distance to cusps") which is $\Gamma_K$-invariant, and can be expressed as $$ y = \max_{\sigma \in \mathbb P^1(K)} y_\sigma $$ where $y_\sigma$ has an explicit expression (I think the best way to describe it is in an adelic setting, especially when there are multiple cusps, but it is clear enough in vdG).

• This yields a partition (almost: the closed sets below have disjoint interior and cover $X$) into subsets $$ F_\sigma = \{ z: y(z) = y_\sigma(z) \} $$ whose stabiliser is exactly $\Gamma_\sigma=\{\gamma\in\Gamma_K:\gamma\sigma=\sigma\}$ (the cusp stabiliser).

• Assume there is only one cusp: then $\Gamma$ permutes the $F_\sigma$ transitively, and it follows immediately that a fundamental domain $D_\sigma$ for $\Gamma_\sigma$ inside $F_\sigma$ is a fundamental domain for $\Gamma$ in $X$.

• The technical part is now to check that the volume of $D_\sigma$ is finite: one way to do this is to observe that it is contained in a subset of the form $\{ y_\sigma \ge s \}$, and the volume of the quotient of such by $\Gamma_\sigma$ is given by integrating something like $\mathrm{vol}(\Omega) dy/y^2$ for $s \le y < +\infty$, where $\Omega$ is a compact fundamental domain for $\Gamma_\sigma$ on some "horosphere" $\{y_\sigma=y_0\}$.

• In the case above the boundary $\partial D_\sigma$ consists of two parts: one inside $F_\sigma$, which looks like $\partial\Omega \times [s,+\infty[$, and the intersection of $\partial F_\sigma$ with $\Omega \times \mathbb R$.

• If there are multiple cusps then one just has to take a $F_\sigma$ for a collection of representatives of the action on $\mathbb P^1(K)$. The "it is easy to see..." remark in vdG seems only to refer to the fact that one may choose these $F_\sigma$s so that their union remains connected.

(I also wanted to point out that another construction of a fundamental domain for the Hilbert--Blumenthal groups has been given By Tamagawa, On Hilbert's modular groups, J. Math. Soc. Japan 1959 no. 3).