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Category: groups

The Big Picture is non-commutative

Conway’s Big Picture consists of all pairs of rational numbers M,gh with M>0 and 0gh<1 with (g,h)=1. Recall from last time that M,gh stands for the lattice
Z(Me1+ghe2)Ze2Q2
and we associate to it the rational 2×2 matrix
αM,gh=[Mgh01]

If M is a natural number we write Mgh and call the corresponding lattice number-like, if g=0 we drop the zero and write M.

The Big Picture carries a wealth of structures. Today, we will see that it can be factored as the product of Bruhat-Tits buildings for GL2(Qp), over all prime numbers p.

Here’s the factor-building for p=2, which is a 3-valent tree:

To see this, define the distance between lattices to be
d(M,gh | N,ij)=log Det(q(αM,gh.αN,ij1))
where q is the smallest strictly positive rational number such that q(αM,gh.αN,ij1)GL2(Z).

We turn the Big Picture into a (coloured) graph by drawing an edge (of colour p, for p a prime number) between any two lattices distanced by log(p).

Misplaced &

The p-coloured subgraph is p+1-valent.

The p-neighbours of the lattice 1=Ze1Ze2 are precisely these p+1 lattices:

pand1p,kpfor0k<p And, multiplying the corresponding matrices with αM,gh tells us that the p-neighbours of M,gh are then these p+1 lattices: pM,pgh mod 1andMp,1p(gh+k) mod 1for0k<p Here's part of the 2-coloured neighbourhood of 1

To check that the p-coloured subgraph is indeed the Bruhat-Tits building of GL2(Qp) it remains to see that it is a tree.

For this it is best to introduce p+1 operators on lattices

pandkpfor0k<p defined by left-multiplying αM,gh by the matrices [p001]and[1pkp01]for0k<p The lattice pM,gh lies closer to 1 than M,gh (unless M,gh=M is a number) whereas the lattices kpM,gh lie further, so it suffices to show that the p operators 0p, 1p,  ,p1p form a free non-commutative monoid.
This follows from the fact that the operator
(knp)(k2p)(k1p)
is given by left-multiplication with the matrix
[1pnk1pn+k2pn1++knp01]
which determines the order in which the ki occur.

A lattice at distance nlog(p) from 1 can be uniquely written as
(knlp)(kl+1p)(pl)1
which gives us the unique path to it from 1.

The Big Picture itself is then the product of these Bruhat-Tits trees over all prime numbers p. Decomposing the distance from M,gh to 1 as
d(M,gh | 1)=n1log(p1)++nklog(pk)
will then allow us to find minimal paths from 1 to M,gh.

But we should be careful in drawing 2-dimensional cells (or higher dimensional ones) in this ‘product’ of trees as the operators
kpandlq
for different primes p and q do not commute, in general. The composition
(kp)(lq)with matrix[1pqkq+lpq01]
has as numerator in the upper-right corner 0kq+l<pq and this number can be uniquely(!) written as kq+l=up+vwith0u<q, 0v<p That is, there are unique operators uq and vp such that (kp)(lq)=(uq)(vp) which determine the 2-cells Misplaced & These give us the commutation relations between the free monoids of operators corresponding to different primes.
For the primes 2 and 3, relevant in the description of the Moonshine Picture, the commutation relations are

(02)(03)=(03)(02),(02)(13)=(03)(12),(02)(23)=(13)(02)

(12)(03)=(13)(12),(12)(13)=(23)(02),(12)(23)=(23)(12)

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The defining property of 24

From Wikipedia on 24:

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/nZ is a square root of 1. It follows that the multiplicative group (Z/24Z)={±1,±5,±7,±11} is isomorphic to the additive group (Z/2Z)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.y1 (mod h) implies xy (mod 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/hZ 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/16Z.

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

A generator of the cyclic group (Z/pZ) is a unit of order p1>2 in Z/pZ, for any prime number p5.

This only leaves those h dividing 23.3=24.

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

Moonshine assigns to elements of the Monster group M a specific subgroup of SL2(Q) containing a cofinite congruence subgroup

Γ0(N)={[abcNd] | a,b,c,dZ,adNbc=1}

for some natural number N=h.n where n is the order of the monster-element, h2 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 Mgh the matrix αMgh=[Mgh01] We say that Γ0(N) fixes Mgh if we have that
αMghΓ0(N)αMgh1SL2(Z)

For those in the know, Mgh stands for the 2-dimensional integral lattice
Z(Me1+ghe2)Ze2
and the condition tells that Γ0(N) preserves this lattice under base-change (right-multiplication).

In “Understanding groups like Γ0(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 Mgh fixed by Γ0(N).

αMgh.[1101].αMgh1=[1M01]

so we must have that M is a natural number, or that Mgh is a number-like lattice, in Conway-speak.

αMgh.[10N1].αMgh1=[1+NgMhNg2Mh2NM1NgMh]

so M divides N, Mh divides Ng and Mh2 divides Ng2. As g and h are coprime it follows that Mh2 must divide N.

Now, for an arbitrary element of Γ0(N) we have

αMgh.[abcNd].αMgh1=[a+cNgMhMbcNg2Mh2(ad)ghcNMdcNgMh]
and using our divisibility requirements it follows that this matrix belongs to SL2(Z) if ad is divisible by h, that is if ad (mod h).

We know that adNbc=1 and that h divides N, so a.d1 (mod h), which implies ad (mod h) if h satisfies the defining property of 24, that is, if h divides 24.

Concluding, Γ0(N) preserves exactly those lattices Mgh for which
1 | M | Nh2  and  h | 24

A first step towards figuring out the Moonshine Picture.

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Pariah moonshine and math-writing

Getting mathematics into Nature (the journal) is next to impossible. Ask David Mumford and John Tate about it.

Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

CAPTION

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

CAPTION

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

Update (October 21st)

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

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