# Current interest in geometric properties of Hilbert fundamental domains

Harvey Cohn published several articles in the 1960's analyzing geometric properties of fundamental domains for Hilbert modular surfaces.

• H. Cohn, "On the shape of the fundamental domain of the Hilbert modular group," Theory of Numbers, A. L. Whiteman (Ed.), Proc. Svmpos. Pure Math. Vol. 8, Amer. Math. Soc, Providence, R. I., 1965,pp. 190-202.
• Cohn, Harvey. "A numerical survey of the floors of various Hilbert fundamental domains." Mathematics of Computation 19.92 (1965): 594-605.
• Cohn, Harvey. "Note on how Hilbert modular domains become increasingly complicated." Journal of Mathematical Analysis and Applications 15.1 (1966): 55-59.
• Cohn, Harvey. "Some computer-assisted topological models of Hilbert fundamental domains." Mathematics of Computation 23.107 (1969): 475-487.

To clarify, he's studying certain fundamental domains for the quotient $$(\mathcal{H}^2\times\mathcal{H}^2)/\mathrm{SL}_2(\mathbb{Z}_K)$$ where $\mathcal{H}^2$ is the upper half-plane, $\mathbb{Z}_K$ is the ring of integers of a real quadratic field $K$, and the group action is by Möbius transformations with a Galois twist. These articles are concerned with nailing down precise geometric properties of the fundamental domains. Especially, he talks about a $3$-manifold boundary for the domain (as a subset of $\mathcal{H}^2\times\mathcal{H}^2$) that arrises in a natural way, which he calls the floor of the domain.

The results mostly talk about the complexity of the regions, some obstacles to describing them in general, and eventually move on to computer generated approximations about specific examples. For instance, the main theorem in the second article above is that $\mathbb{Q}(\sqrt 5)$ is the only UFD for which the floor has just one piece. The fourth article above crunches some data and some (circa 1969) computer generated images of cross-sections of floors for the cases $\mathbb{Q}(\sqrt n)$ with $n=2,3,5,6$.

My current research project has lead me to a result that allows me to say a lot more about what's happening here. (This wasn't really the plan, as research goes, but to me it is very interesting especially when I look back at what Cohn did.) I'm wondering if (or how much) these results might interest the community. There is a lot already known about these objects' arithmetic and topological properties, and what I have (so far) does not extend these. It does give a different way of seeing/explaining some known facts in a hands-on way, which is cool, but that's not exactly Theorem-(with a capital T)-worthy.

What I do have regards putting precise shapes to the objects, giving algorithms to compute sides, showing models for 3D cross-sections (in circa 2017 computer graphics -- interestingly I think Cohn would have noticed some stuff I noticed if he just had better resolution!) ... things like that. But I don't have a good sense of the relevance of these properties to the current study of Hilbert modular surfaces. So, who (if anyone) might find that kind of thing interesting, and why? Would it make a decent paper on its own? Can you direct me to some other relevant literature? Thanks in advance!

• I hope you don't mind, I repaired two typos that tweaked my neuroses... :) – paul garrett Oct 19 '17 at 23:26
• @paulgarrett Not at all! I appreciate very much your attention to my questions, you seem to be knowledgeable about a lot of the things I'm interested in! – j0equ1nn Oct 26 '17 at 0:20
• Alberto Verjovsky and I have a paper on the results mentioned in the question, you can find it here: arxiv.org/abs/1711.02418 – j0equ1nn Nov 16 '17 at 17:11
• Ah, yes, I saw that paper! Some non-trivial results! – paul garrett Nov 16 '17 at 18:14

I make the disclaimer that I am by no means an "expert" on computational aspects of modular forms, although I am aware of some of the issues.

First, I'd think you should look at the Sage (sagemath.org) situation, where they already have a great many algorithmic things packaged-up. Many elliptic modular forms things are automated, for example. Graphs of zeta at various perspectives are nearly instantaneous.

Some decades ago, there were some delicate results in elliptic modular forms exploiting subtle details about choices of fundamental domains... but, I'm very sorry, I do not remember the authors or the results.

I should say that it is actually lucky that most basic results about Hilbert modular forms or other groups' automorphic forms do not depend much on details about fundamental domains, since such details are irregular and expensive. So there'd not be an immediate, general benefit, I think, in knowing something about fundamental domains.

For that matter, there have been theorems proven indicating that in higher rank, generic Shimura varieties are "of general type" as algebraic varieties, so nothing "sharp" can be said from that side. Likewise, this might cause one to doubt that details about a fundamental domain could be broadly helpful. But perhaps I am mistaken.

(Also, there was a paper by Hammond in the Boulder conference about explicit generation of rings of Hilbert modular forms.)

• Well, I was able to make fundamental domains with simple floors (in the sense of Cohn) for examples beyond $n=5$, the only one where this happens in the (naive) classical construction. I'm also able to connect floor simplicity to some interesting arithmetic properties that seem suggestive of other applications. Whether it actually advances the state of the art may be unclear but at the very least it's a different angle on the idea that makes it accessible to additional areas of math. So... I'm going to write it up and submit it after all. – j0equ1nn Oct 25 '17 at 2:42
• I feel I may be able to do more with it, but I also feel like it's important to represent in the article what it was that actually interested me and be able to focus on that. After all if it turns out that my interests aren't those of the broader community, I'd rather bark up a different (mathematical) tree. – j0equ1nn Oct 25 '17 at 2:45
• I have a follow-up question about another application that I can easily get with my new fundamental domain. It's something I can include in a couple extra sentences but for the life of me I cannot determine if it was previously known. It has to do with finding the Anosov diffeomorphism of a cusp section. It is known that such a diffeomorphism exists, but has anyone been able to specify it? – j0equ1nn Oct 26 '17 at 0:22