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The defining property of 24

Published January 2, 2018 by lieven

“ $24$ is the only number whose divisors, namely $1, 2, 3, 4, 6, 8, 12, 24$ , are exactly those numbers $n$ for which every invertible element of the commutative ring $Z / n Z$ is a square root of $1$ . It follows that the multiplicative group $(Z / 24 Z)^{*} = {\pm 1, \pm 5, \pm 7, \pm 11}$ is isomorphic to the additive group $(Z / 2 Z)^{3}$ . This fact plays a role in monstrous moonshine.”

Where did that come from?

In the original “Monstrous Moonshine” paper by John Conway and Simon Norton, section 3 starts with:

“It is a curious fact that the divisors $h$ of $24$ are precisely those numbers $h$ for which $x . y \equiv 1 (m o d h)$ implies $x \equiv y (m o d h)$ .”

and a bit further they even call this fact:

“our ‘defining property of $24$ '”.

The proof is pretty straightforward.

We want all $h$ such that every unit in $Z / h Z$ has order two.

By the Chinese remainder theorem we only have to check this for prime powers dividing $h$ .

$5$ is a unit of order $4$ in $Z / 16 Z$ .

$2$ is a unit of order $6$ in $Z / 9 Z$ .

A generator of the cyclic group $(Z / p Z)^{*}$ is a unit of order $p - 1 > 2$ in $Z / p Z$ , for any prime number $p \geq 5$ .

This only leaves those $h$ dividing $2^{3} .3 = 24$ .

But, what does it have to do with monstrous moonshine?

Moonshine assigns to elements of the Monster group $M$ a specific subgroup of $S L_{2} (Q)$ containing a cofinite congruence subgroup

$Γ_{0} (N) = {[\begin{matrix} a & b \\ c N & d \end{matrix}] | a, b, c, d \in Z, a d - N b c = 1}$

for some natural number $N = h . n$ where $n$ is the order of the monster-element, $h^{2}$ divides $N$ and … $h$ is a divisor of $24$ .

To begin to understand how the defining property of $24$ is relevant in this, take any strictly positive rational number $M$ and any pair of coprime natural numbers $g < h$ and associate to $M \frac{g}{h}$ the matrix $α_{M \frac{g}{h}} = [\begin{matrix} M & \frac{g}{h} \\ 0 & 1 \end{matrix}]$ We say that $Γ_{0} (N)$ fixes $M \frac{g}{h}$ if we have that
$α_{M \frac{g}{h}} Γ_{0} (N) α_{M \frac{g}{h}}^{- 1} \subset S L_{2} (Z)$

For those in the know, $M \frac{g}{h}$ stands for the $2$ -dimensional integral lattice
$Z (M {\vec{e}}_{1} + \frac{g}{h} {\vec{e}}_{2}) \oplus Z {\vec{e}}_{2}$
and the condition tells that $Γ_{0} (N)$ preserves this lattice under base-change (right-multiplication).

In “Understanding groups like $Γ_{0} (N)$ ” Conway describes the groups appearing in monstrous moonshine as preserving specific finite sets of these lattices.

For this, it is crucial to determine all $M \frac{g}{h}$ fixed by $Γ_{0} (N)$ .

$α_{M \frac{g}{h}} . [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] . α_{M \frac{g}{h}}^{- 1} = [\begin{matrix} 1 & M \\ 0 & 1 \end{matrix}]$

so we must have that $M$ is a natural number, or that $M \frac{g}{h}$ is a number-like lattice, in Conway-speak.

$α_{M \frac{g}{h}} . [\begin{matrix} 1 & 0 \\ N & 1 \end{matrix}] . α_{M \frac{g}{h}}^{- 1} = [\begin{matrix} 1 + \frac{N g}{M h} & - \frac{N g^{2}}{M h^{2}} \\ \frac{N}{M} & 1 - \frac{N g}{M h} \end{matrix}]$

so $M$ divides $N$ , $M h$ divides $N g$ and $M h^{2}$ divides $N g^{2}$ . As $g$ and $h$ are coprime it follows that $M h^{2}$ must divide $N$ .

Now, for an arbitrary element of $Γ_{0} (N)$ we have

$α_{M \frac{g}{h}} . [\begin{matrix} a & b \\ c N & d \end{matrix}] . α_{M \frac{g}{h}}^{- 1} = [\begin{matrix} a + c \frac{N g}{M h} & M b - c \frac{N g^{2}}{M h^{2}} - (a - d) \frac{g}{h} \\ c \frac{N}{M} & d - c \frac{N g}{M h} \end{matrix}]$
and using our divisibility requirements it follows that this matrix belongs to $S L_{2} (Z)$ if $a - d$ is divisible by $h$ , that is if $a \equiv d (m o d h)$ .

We know that $a d - N b c = 1$ and that $h$ divides $N$ , so $a . d \equiv 1 (m o d h)$ , which implies $a \equiv d (m o d h)$ if $h$ satisfies the defining property of $24$ , that is, if $h$ divides $24$ .

Concluding, $Γ_{0} (N)$ preserves exactly those lattices $M \frac{g}{h}$ for which
$1 | M | \frac{N}{h^{2}} and h | 24$

A first step towards figuring out the Moonshine Picture.

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Monsters and Moonshine : a booklet

Published January 19, 2012 by lieven

I’ve LaTeXed $48 = 2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.

The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$ -sliding game which uses the sporadic Mathieu group $M_{12}$ . This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’. We discuss Galois’ last letter and the simple groups $L_{2} (5), L_{2} (7)$ and $L_{2} (11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered ‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover what the largest sporadic simple group, the Monster-group, sees of the modular group.

The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $P S L_{2} (Z)$ . We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using Farey codes and use those to give a series of simple groups including as special members $L_{2} (5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E (8)$ in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s solution of the $E (8)$ -observation.

I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!

If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to the “field with one element” and its connections to (noncommutative) geometry and number theory.

I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so far only exists as prep-notes. The links above will always link to the latest versions of these booklets.

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the Reddit (after)effect

Published January 15, 2011 by lieven

Sunday january 2nd around 18hr NeB-stats went crazy.

Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day.

Now, the dust has settled, so let’s learn from the experience.

A Reddit-mention is to a blog what doping is to a sporter.

You get an immediate boost in the most competitive of all blog-stats, the number of unique vistors (blue graph), but is doesn’t result in a long-term effect, and, it may even be harmful to more essential blog-stats, such as the average time visitors spend on your site (yellow graph).

For NeB the unique vistors/day fluctuate normally around 300, but peaked to 1295 and 1733 on the ‘Reddit-days’. In contrast, the avg. time on site is normally around 3 minutes, but dropped the same days to 44 and 30 seconds!

Whereas some of the Reddits spend enough time to read the post and comment on it, the vast majority zap from one link to the next. Having monitored the Reddit/math page for two weeks, I’m convinced that post only made it because it was visually pretty good. The average Reddit/math-er is a viewer more than a reader…

So, should I go for shorter, snappier, more visual posts?

Let’s compare Reddits to those coming from the three sites giving NeB most referrals : Google search, MathOverflow and Wikipedia.

This is the traffic coming from Reddit/math, as always the blue graph are the unique visitors, the yellow graph their average time on site, blue-scales to the left, yellow-scales to the right.

Here’s the same graph for Google search. The unique visitors/day fluctuate around 50 and their average time on site about 2 minutes.

The math-related search terms most used were this month : ‘functor of point approach’, ‘profinite integers’ and ‘bost-connes sytem’.

More rewarding to me are referrals from MathOverflow.

The number of visitors depends on whether the MathO-questions made it to the front-page (for example, the 80 visits on december 15, came from the What are dessins d’enfants?-topic getting an extra comment that very day, and having two references to NeB-posts : The best rejected proposal ever and Klein’s dessins d’enfant and the buckyball), but even older MathO-topics give a few referrals a day, and these people sure take their time reading the posts (+ 5 minutes).

Other MathO-topics giving referrals this month were Most intricate and most beautiful structures in mathematics (linking to Looking for F-un), What should be learned in a first serious schemes course? (linking to Mumford’s treasure map (btw. one of the most visited NeB-posts ever)), How much of scheme theory can you visualize? (linking again to Mumford’s treasure map) and Approaches to Riemann hypothesis using methods outside number theory (linking to the Bost-Connes series).

Finally, there’s Wikipedia

giving 5 to 10 referrals a day, with a pretty good time-on-site average (around 4 minutes, peaking to 12 minutes). It is rewarding to see NeB-posts referred to in as diverse Wikipedia-topics as ‘Fifteen puzzle’, ‘Field with one element’, ‘Evariste Galois’, ‘ADE classification’, ‘Monster group’, ‘Arithmetic topology’, ‘Dessin d’enfant’, ‘Groupoid’, ‘Belyi’s theorem’, ‘Modular group’, ‘Cubic surface’, ‘Esquisse d’un programme’, ‘N-puzzle’, ‘Shabat polynomial’ and ‘Mathieu group’.

What lesson should be learned from all this data? Should I go for shorter, snappier and more visual posts, or should I focus on the small group of visitors taking their time reading through a longer post, and don’t care about the appallingly high bounce rate the others cause?

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Tag: monster

The defining property of 24

Monsters and Moonshine : a booklet

the Reddit (after)effect