The n-Category Café

So that William doesn’t feel too bad, I will add that his understanding corresponds to mine (right down to the holes poked in the surface). Of course, I’m just another amateur, so what I’m about to say may very well be completely idiotic, but the urge to babble has become irresistable, so here goes.

My understanding (if you can call it that) is that the inclusion of $\Gamma (N)$ in the modular group $\Gamma$ gives rise to a covering of $H/\Gamma$ by $H/\Gamma (N)$, and lifting the fundamental domain of $\Gamma$ to the covering gives a tiling of the surface by regular $N$-gons, 3 meeting at each vertex, each subdivided into $N$ triangles corresponding to the fundamental domain of the modular group, From this, it ought to be obvious that the cusps correspond to the centres of the faces. So if that’s not true, then something has going terribly wrong …

I thought that, except for the case of level 2, there are $\frac{N^2}{2}{\prod }_{p\mid N}(1-\frac{1}{p^2})$ N-gons (taking the product over all prime factors $p$ of $N$, each counted once. (In fact I’m sure I read this, or something like it, in Diamond and Shurman, only of course now I can’t find it.) This does agree with what William said about the Platonic solids. The faces can, I believe, be labelled by lines through the origin of the free 2-dimensional module over ${\mathbb{Z}}_N$, although in general, these labels will appear on multiple faces. (For prime $N$, these will be points in the projective line over the field with $N$ elements; otherwise they will be somewhat wackier points in a generalised “projective line” over a ring. E.g. for $N=6$ we get points $0$, $1$, $2$, $3$, $4$, $5$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{3}{4}$ and $\mathrm{\infty }$.)

However, the action of $\Gamma /\Gamma (N)$ will be consistent over where it sends the labels. So, for instance, the 24 faces of the Klein quartic can be labelled with the 8 points in the projective line over ${𝔽}_7$, each appearing three times, and if a member of $\mathrm{SL}(\mathrm{2,7})$ sends, say one face labelled with 2 to a face labelled with $\mathrm{\infty }$, then it will send all three faces labelled with 2 to faces labelled with $\mathrm{\infty }$.

The modular group will then act as automorphisms of the tiling. The generator sending $z\to z+1$ will rotate the faces labelled with $\mathrm{\infty }$ by one step (ie one Nth of a full rotation), the generator sending $z\to -\frac{1}{z}$ will rotate 180 degrees about the edge(s) connecting the face(s) labelled with $\mathrm{\infty }$ and the face(s) labelled with $0$, and likewise, order three elements rotate about vertices.

As William said, I thought at level 2, we get the tiling of the Riemann sphere with regular bigons, three meeting at each vertex, giving 3 faces (labelled with 1, 0 and $\mathrm{\infty }$), 2 vertices and 3 edges.

Or perhaps this is all baloney.

Hmm, looking at what you said again, you’re talking not about $\Gamma (2)$, but about ${\Gamma }_0(2)$. That might account for the confusion. At least, if it did, I am less likely to be humiliated and equally likely to learn something when a genuine professional turns up to explain things. So I’m keeping my fingers crossed.

Yes, that is certainly is one of the most fascinating themes in mathematics. Esp. the use of modular functions to build class fields in analogy to the use of the exponential function to produce roots of unity. Here a nice article. I found this book on that very thrilling. One of the IMHO most astonishing things is the existence of analoga over finite fields, e.g. here the “algebraist’s upper half plane”, related to other strange things. A beautifull introduction into such developments is the last chapter of Vladut “Kronecker’s Jugendtraum and Modular Functions”. Finally a related ppt and here and here articles about the strange conections between modular functions and continued fractions and Y.I. Manins “Alterstraum”.

And yes, I meant level-two rather than level-one. I meant weight-one or weight-two there, anyways.

OK, got a little more.

The labels on the cusps of $\Gamma (N)$ are precisely those members of the group ${\mathbb{Z}}_N^2$ which are of order $N$, modulo negation. This means that we get a labelling by lines through the origin of the ${\mathbb{Z}}_N$-module ${\mathbb{Z}}_N^2$, each of which appears a number of times equal to half the order of the group of units in ${\mathbb{Z}}_N$. So twice in the case of $\Gamma (5)$, three times in the case of $\Gamma (7)$, once for $\Gamma (3)$, $\Gamma (4)$ and $\Gamma (6)$. ($\Gamma (2)$ is slightly different.)

Now, in the case of ${\Gamma }_1(N)$, we mod out by the action of the group generated by $z\to z+1$. So the $N$-gon(s) labelled by $\mathrm{\infty }$ fold down to a single triangle made into a cone, with the cusp at its apex. (The triangles, I neglected to mention, have bases on the edges of the $N$-gon, and apices at its centre.) The faces labelled with integers all collapse into a single $N$-gon. Other faces (if any) may undergo various other forms of collapse.

In the case of ${\Gamma }_1(2)$, the faces labelled with $0$ and $1$ become identified, so we get left with 2 cusps, one at $0$ (or $1$) and one at $\mathrm{\infty }$.

In the case of, say, ${\Gamma }_1(4)$, we get a cusp on the $\mathrm{\infty }$ face, a cusp on the “integer” face (which results from identifying 0, 1, 2 and 3) and a cusp on the $\frac{1}{2}$ face.

Some of these cusps (not to mention the other elliptic points) get folded over in various ways which probably do things do their order that I haven’t worked out yet.

I still haven’t understood ${\Gamma }_0(N)$. It’s obviously somewhere in between the other two, but it seems to be more complicated than either.

JB wrote:

PSL(2,5) is the symmetry group of the dodecahedron, and PSL(2,7) is the symmetry group of the Klein quartic

Yes, and PSL(2,3) is the symmetry group of the tetrahedron ($A_4$) and PSL(2,2) is the symmetry group of the tiling of the sphere by three bigons ($S_3$) and the group that one might call $\mathrm{PSL}(2,{\mathbb{Z}}_4)$ (linear transformations of ${\mathbb{Z}}_N\times {\mathbb{Z}}_N$ of determinant 1, mod $\{-\mathrm{1,1}\}$) is the symmetry group of the cube ($S_4$).

(Is there any deep reason for that progression of symmetric and alternating groups $S_3$, $A_4$, $S_4$, $A_5$, other than numerical coincidence?)

All these comments are great! I’m supposed to be writing a paper, so I’m trying not to blog too much. I will read these more carefully and reply when I feel like taking a break and having some fun.

The main next things I’d like to do are:

• Really understand the roles of $\mathrm{SL}(2,\mathbb{Z}/N)$ versus $\mathrm{PSL}(2,\mathbb{Z}/N)$ in this situation, at least when $N$ is prime. And do they make sense when $N$ isn’t prime?
• At least for $N=p$ an odd prime, see why the $p^3-p$ triangles tiling $H/\Gamma (N)$ organize themselves into $p$-gons with $2p$ triangles each, for a total of $\frac{1}{2}(p^2-1)$ $p$-gons.
• Figure out how many of these $p$-gons meet at each vertex. It seems to be 3 a lot of the time. Is it always 3?
• If it’s always 3 $p$-gons meeting at a vertex, I can calculate the genus of $H/\Gamma (p)$, and see why we’re getting genus 0 (a sphere) for $p=3$ and $5$, but genus 3 (a 3-holed torus, shown writhing above) for $p=5$.
• It would then be fun to understand $p=11$ — some sort of surface tiled by 120 hendecagons.
  
• Of course, non-prime $N$ are also interesting.

JB wrote

I’m way behind you

That’s a slightly surreal and very scary position for me to find myself in. Some of this is stuff I think I actually know, but lots of it is just strung together with connections that seem intuitively plausible, backed up by some numbers and groups I’ve calculated that turn out to match the answers in the books…

That said, let me make a couple of things clear which might have confused some of our readers. First, I’m only going to talk about $\Gamma (n)$, which I think I sort of understand, and not ${\Gamma }_0(N)$ or ${\Gamma }_1(N)$, which I’m much less clear about. Second, it’s as well to be explicit that there are two kinds of triangle that we are talking about. We can, for instance, take the fundamental domain of the modular group—the grey triangle in the first diagram above. Or we can slice it vertically up the middle to get two triangles—that’s also the kind of triangle in the second diagram, the one where the triangles are coloured. Obviously there are twice as many of the second type of triangle, so we need to be clear which kind we’re talking about. In my earlier posts, I was always talking about the first kind.

So that gives us the number of elements in $\mathrm{PSL}(2,N)$, which is what I want. And yes, it does appear, from my not very rigorous or thorough investigations, that this can be made to work for non-prime $N$. We just have to think carefully about the points in ${\mathbb{Z}}_N^2$, and the effect on them of matrices taking values in ${\mathbb{Z}}_N$, and multiplication by scalars in ${\mathbb{Z}}_N$.

Darn. There’s plenty more I wanted to say, but it’s getting late, so maybe I’ll talk more tomorrow. In the meantime, I can refer you back to §3.8 of Diamond and Shurman, which is full of excellent stuff.

John wrote:

Knapp seems to focus on the groups ${\Gamma }_0(n)$. This, and the fact that they’re called “Hecke subgroups”, seems to suggest they’re especially important.

Jim Dolan replied via email:

is the structure on a lattice that the hecke subgroup stabilizes (say for prime n) essentially just an “n-local flag” or something like that? (in this dimension “flag” is just a bombastic way of saying “non-trivial subspace”, right?)

i didn’t actually try to calculate this or check any fine print or anything.

That sounds right, since this group consists of “upper triangular matrices mod n”, and in general upper triangular matrices preserve a flag.

Jim wrote in reply to the above stuff:

whenever you have a “basic hecke operator” (aka “atomic geometric relationship”), there’s the “joint stabilizer” subgroup simultaneously stabilizing both of the features involved in the relationship; aka “intersection of the original stabilizers in the given relative position”. might this be what a “hecke subgroup” is, in general? so in this particular case, i’m imagining that maybe there’s a basic hecke operator something like “jump to an n-fold refinement of the lattice” (thought of as a random jump operator on lattices), and that the intersection of the two different sl(2,z)’s respectively stabilizing the original lattice and its n-fold refinement might be a “gamma_0(n)” or whatever it’s called; can you get something like that to work?

if this _is_ (more or less) what “hecke subgroups” are, then i’m a bit annoyed that i didn’t hear anyone say it outloud before. on the other hand the guess that i’m making here _is_ influenced by stuff gleaned from brown’s book, about the incidence geometry corresponding to the circular “augmented a_n series” dynkin diagrams.

Yup. You’ve kind of followed the same reasoning that I used when working this stuff out, only backwards. I started out thinking, Well, mod $N$, that means the operation of adding $1$ should wrap round on itself after $N$ steps … and went on from there.

Maybe the guy who told you about the one cusp was thinking of the one cusp at $\mathrm{\infty }$ which lifts to multiple cusps when we go to mod $N$.

Tim wrote:

Maybe the guy who told you about the one cusp was thinking of the one cusp at $\mathrm{\infty }$ which lifts to multiple cusps when we go to mod $N$.

Maybe that’s it. Let’s say that’s it.

(This is the problem with ‘learning math on the streets’…. rumors spread and get a bit more distorted with each retelling.)

If I weren’t so busy right now, I’d post another entry on modular forms. I’ve made a lot of progress, mainly thanks to talking with Jim. I guess I’ll talk about it when I get time.

One cool fact: the Modularity Theorem says each rational elliptic curve is the moduli space for elliptic curves equipped with some sort of extra structure. Very conceptual and self-referential! Amazing that Fermat’s Last Theorem is a corollary.

The p = 11 case is in the literature after all – by Chemists!

“The undecakisicosahedral group and a 3-regular carbon network of genus 26”, by Erwin Lijnen, Arnout Ceulemans, Patrick W. Fowler, and Michel Deza

Abstract:

Three projective special linear groups PSL(2,p), those with p = 5, 7 and 11, can be seen as p-multiples of tetrahedral, octahedral and icosahedral rotational point groups, respectively. The first two have already found applications in carbon chemistry and physics, as PSL(2,5) ≡ I is the rotation group of the fullerene C_60 and dodecahedrane C_20H_20, and PSL(2,7) is the rotation group of the 56-vertex all-heptagon Klein map, an idealisation of the hypothetical genus-3 “plumber’s nightmare” allotrope of carbon.
Here, we present an analysis of PSL(2,11) as the rotation group of a 220-vertex, all 11-gon, 3-regular map, which provides the basis for a more exotic hypothetical sp^2 frame-work of genus 26. The group structure and character table of PSL(2,11) are developed in chemical notation and a three dimensional (3D) geometrical realisation of the 220-vertex map is derived in terms of a punctured polyhedron model where each of 12 pentagons of the truncated icosahedron is connected by a tunnel to an interior void and the 20 hexagons are connected tetrahedrally in sets of 4.
KEY WORDS: PSL(2,11), group theory, undecakisicosahedral group, topology, carbon allotrope
AMS: 05C10, 20B25, 57M20

1. Introduction

The three groups PSL(2,5), PSL(2,7) and PSL(2,11) form a special sub-set of the Projective Special Linear groups PSL(2,p) in view of their particular permutational structure. They can be viewed as multiples of the symmetry groups of the regular polyhedra in three dimensional (3D) space, and for this reason are also called the pollakispolyhedral groups [1]. PSL(2,5) and PSL(2,7) correspond to the pentakistetrahedral,
5_T, and heptakisoctahedral group,
7_O, respectively. Both have found applications in chemistry and physics….
[truncated]

Bingo! Does that fit the bill?

Now, can Greg Egan animate this into a screensaver of dazzling brain-twisting gorgeousness?

Thomas Riepe wrote:

Conc. “modular forms and $L$-series”:

When I thought about mentioning the connection between both as esp. interesting, because in Hecke’s Theory the Mellin transform makes the functional eq.s of some $L$-series just the defining transformation rules of associated modular forms, I noticed that I know nothing about the $p$-adic analogon e.g. of the Mellin transform. Perhaps someone here knows where to look?

I’m no expert, but at my current stage of development I find it reassuring to regard the Mellin transform as a silly trick — part of a cute but distracting ‘explicit formula’ for the transform mapping the Taylor series

$$\sum _{n\ge 1}{a}_n{z}^n$$

to the Dirichlet series

$$\sum _{n\ge 1}{a}_n{n}^z$$

Here’s how I’d prefer to think about it:

$ℂ$-valued functions on a sufficiently well-behaved monoid $M$ form an algebra.

The algebra of Taylor series is the algebra of complex functions on the additive monoid of natural numbers.

The algebra of Dirichlet series is the algebra of complex functions on the multiplicative monoid of nonzero natural numbers.

Since these two monoids are both part of a single structure — the rig of natural numbers, the algebras of Taylor and Dirichlet series are part of a single algebraic structure: essentially, a ‘rig algebra’.

But, since most people have never thought about rig algebras, they jump back and forth between the two pieces of the rig algebra of $\mathbb{N}$ with the help of the Mellin transform and its inverse.

Unfortunately this brings analysis into what is at its core a simple algebraic idea which would make sense with any field (e.g. the $p$-adics) replacing $ℂ$.

I’m overstating my case here, because at certain points the analytic aspects of the relation between modular forms and $L$-functions do become important — e.g. when we gain information about algebraic gadgets from poles and zeroes of $L$-functions that we can cook up from them. At least, that’s what everyone keeps saying.

But, I still find it amusing that people seem loathe to simply define a transform by saying it sends

$$\sum _{n\ge 1}{a}_n{z}^n$$

to

$$\sum _{n\ge 1}{a}_n{n}^z$$

and worry later about whether there’s an integral formula for this transform. After all, as soon as you get integrals into the game you start having to worry about when the integrals converge. I’d prefer to worry about that only when it becomes crucial.

Of course, when I learn more about this I may decide my current viewpoint is sort of childish.

(By the way, in the stuff above, a monoid is ‘sufficiently nice’ if for every element $x$ there are finitely many pairs $y,z$ with $yz=x$. We could avoid this issue by working with the monoid algebra $ℂ[M]$, which you can think of as consisting of functions on $M$ that vanish except at finitely many points. However, the Dirichlet series appearing in number theory usually have infinitely many nonzero terms, so I’m not taking that approach.)

” Here the very readable review article of Katz on Delignes proof of the Weil conjectures…”

Oops, I meant this article. From the MathSciNet-review:

“This paper makes accessible to non-specialists the principal ideas in P. Deligne’s proof of the following fundamental result: …

The author begins with an historical and motivational survey of this Riemann hypothesis, explaining the connection with formulae and inequalities for the number of solutions of equations over finite fields, the ideas behind the early proofs of special cases, the historical role of the Riemann hypothesis and the other Weil conjectures in the development of algebraic geometry, and the heuristic argument for the conjectures based on classical (characteristic 0) phenomena.

Concerning Deligne’s proof itself, the author spotlights two ingredients—(1) the monodromy of Lefschetz pencils, and (2) modular forms and Rankin’s method for studying the Ramanujan function and describes their fundamental roles in the proof. Deligne’s proof is then outlined in more detail in the special case of odd-dimensional non-singular hypersurfaces. Here monodromy, l-adic cohomology, Rankin’s method and L-series occupy a central place. The author discusses applications to the Ramanujan-Petersson conjecture, estimation of exponential sums in several variables, and the “hard” Lefschetz theorem. He concludes with a discussion
of open questions. The paper includes a lengthy bibliography.”