Let $d\in\mathbb{N}$ be squarefree. Let $\mathcal{O}_d$ be the ring of integers of $\mathbb{Q}(\sqrt{-d})$. Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$. Let $\mathcal{H}^3$ be the upper half-space model for hyperbolic 3-space. Let $X_d=\mathcal{H}^3/\Gamma_d$ be the orbifold obtained by the usual action of $\Gamma_d$ on $\mathcal{H}^3$ (do Mobius transformations on the boundary, and extend them upward isometrically). $X_d$ is called a Bianchi orbifold.

Bianchi was an Italian mathematician who died in 1928. An orbifold is like a manifold but with cone points, and was defined by Thurston in the 1970's. First question: to what extent did Bianchi understand the geometry of $X_d$, and what were his main tools in studying these objects?

Theorem: The number of cusps of $X_d$ is equal to the class number of $\mathbb{Q}(\sqrt{-d})$.

In the literature, this is attributed to Hurwitz in 1892. I have that reference but it is in German (which would take me a long time to translate). Class numbers have been around since Gauss, but my understanding of a cusp (like orbifolds) is from a rather contemporary topological standpoint. We could define a cusp of $X_d$ as a conjugacy class of maximal parabolic subgroups of $\Gamma_d$ but when you say it like that it doesn't sound nearly as interesting. What I love about this theorem is that it's pure number theory on one end and pure topology on the other. So, second question: what was a "cusp" as far as Hurwitz was concerned, and why would he have found them interesting?

  • 4
    $\begingroup$ Just a guess : $PSL_2(\mathcal O_d)$ acts in a natural way on the projective line $P^1(\mathbf Q[\sqrt d])$, and the quotient is in bijection with the class group on the one hand, and (quite trivially) with the cusps on the other hand. Thus maybe Hurwitz understood this first quotient ... $\endgroup$
    – few_reps
    Mar 8, 2016 at 7:09
  • 3
    $\begingroup$ You could ask all the same questions about classical modular curves which historically will be even older. In my modular forms class I was just told to replace $\Gamma$ by something smaller to remove technicalities involving stabilisers (so my teacher didn't need to invent orbifolds) and I was also told that you compactified them by adding $P^1(Q)$ and these were call cusps (so again no topology). I'm not saying that's what Hurwitz was thinking for sure, but I am saying that one can set up the theory in an entirely logically consistent way without the topological insight. $\endgroup$
    – wrigley
    Mar 8, 2016 at 7:22
  • 2
    $\begingroup$ This is an interesting question for MO -- I would prefer it stay here than get migrated elsewhere. $\endgroup$
    – Lucia
    Mar 8, 2016 at 13:37
  • 2
    $\begingroup$ As in @few_reps' comment: Dedekind knew that ideals in rings of algebraic integers need at most two generators, so orbits of $PGL_2(\mathfrak o_k)$ on the projective line over $k$ are in bijection with ideal classes. Replacing $PGL$ by $PSL$ gives narrow ideal classes... $\endgroup$ Mar 8, 2016 at 13:44
  • 3
    $\begingroup$ I wanted to have a look at Hurwitz's paper, but could not track down a reference. The only published paper in 1892 seems to be the one where he proves the Hurwitz formula for coverings of Riemann surfaces. What exactly is the Hurwitz reference mentioned in the question? $\endgroup$ Mar 9, 2016 at 14:26

1 Answer 1


It is very difficult to find a paper of Hurwitz dealing in any way with the geometry or topology of $\mathbb{H}^3/PSL_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-D})})$. (And I am not sure I like what this implies about our ways of keeping knowledge alive, attribution and such.) The closest two things I could find were the following:

  1. A paper of Bianchi on computations of fundamental domains for $PSL_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-D})})$:

L. Bianchi. Geometrische Darstellung der Gruppen linearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie. Math. Ann. 38 (1891), 313-333.

The main purpose of the investigation seems to have been reduction theory (and classification) of quadratic forms. The description of the fundamental domain is based on explicit generators for the group. The discussion of the topology is based on Poincaré's hyperbolic space model, the explicit transformation formulas for the group action, and the metric of hyperbolic space.

On page 2 of Bianchi's paper there is a reference to

A. Hurwitz: Über die Entwicklung complexer Grössen in Kettenbrüche. Acta Math. 11 (1888), 187-200.

That paper is mostly about continued fractions and discusses what could possibly be interpreted as related to the fundamental domains for $PSL_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-D})})$ for $D=1$ and $D=3$. It seems, however, to be a rather indirect link of Hurwitz to cusps.

  1. Next, there is a 1892 paper of Bianchi:

L. Bianchi. Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî. Math. Ann. 40 (1892), 332-412.

In this, he states the relevant theorem: "Il numero dei vertici singolari eguaglia il numero delle classi degli ideali nel corpo quadratico corrispondente." Singular vertices for him are orbits of boundary points (or boundary points in each fundamental domain), and it seems to me that in §§2 and 3 he essentially proves this result by computing the orbits of $PSL_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-D})})$ on the boundary projective line (as already suggested in various comments).

There is a reference to Hurwitz's work here as well:

A. Hurwitz. Grundlagen einer independenten Theorie der Modulfunctionen. Math. Ann 18 (1881), pp. 528-592.

For some reason this paper does not seem to be in MathSciNet. Anyway, Bianchi says

"Ma senza riferirci al teorema generale di Poincaré daremo qui una dimostrazione diretta di questa proprietà affatto analoga a quella che il Sig. Hurwitz ha fatto conoscere pel gruppo modulare."

Meaning that Bianchi's computation of the fundamental domain does not make use of Poincaré's general theorem, but is proved directly analogous to Hurwitz's arguments for the modular group. Hurwitz's treatment of fundamental domains for the modular group and its congruence subgroups on the upper half plane (in the abovementioned paper) seems to consider only the upper half plane, and not discuss boundary points.

Conclusion: At least from these two findings, it seems that Hurwitz was not directly involved in the proof that cusps for the Bianchi groups are in bijection with ideal classes. Maybe additional information could be inferred from Hurwitz's mathematical diaries which are available from the ETH library.

It is not clear to me if Bianchi really did consider the topology of the orbifold. He considered the group $PSL_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-D})})$ (or $PGL_2$) as generated by an explicit set of transformations, and he viewed the group action on $\mathbb{H}^3$ in terms of explicit formulas. From the papers above it seems he did not think about an orbit space, but just about the construction of some fundamental polyhedron which contained representatives for all orbits. From this point of view, of course, cusps and the quotient topology are not an issue at all since everything takes place in $\mathbb{H}^3$ or $\overline{\mathbb{H}^3}$. (The same applies to Hurwitz's treatment of the action of the modular group on the upper half plane.)

In any case, in view of the above references it seems to me more appropriate to credit Bianchi (in 1892) with the identification of cusps and ideal classes.

  • $\begingroup$ This investigation is very much appreciated! I think this takes care of question one and probably question two as well. I wonder though if there are any disagreements about Hurwitz's role. Afterall there are plenty of books/papers attributing this to Hurwitz. I'd be especially interested in @AllenHatcher 's take. In his book he is careful about history and only says that this theorem "is known." $\endgroup$
    – j0equ1nn
    Mar 10, 2016 at 13:26
  • 1
    $\begingroup$ Here's a link to Hurwitz's paper: digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002246309 $\endgroup$
    – David Roberts
    May 19, 2019 at 1:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.