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Tag: Norton

a non-commutative Jack Daniels problem

At a seminar at the College de France in 1975, Tits wrote down the order of the monster group

#M=246.320.59.76.112.133.17·19·23·29·31·41·47·59·71

Andrew Ogg, who attended the talk, noticed that the prime divisors are precisely the primes p for which the characteristic p super-singular j-invariants are all defined over Fp.

Here’s Ogg’s paper on this: Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 16, no 1 (1974-1975).

Ogg offered a bottle of Jack Daniels for an explanation of this coincidence.

Even Richard Borcherds didn’t claim the bottle of Jack Daniels, though his proof of the monstrous moonshine conjecture is believed to be the best explanation, at present.

A few years ago, John Duncan and Ken Ono posted a paper “The Jack Daniels Problem”, in which they prove that monstrous moonshine implies that if p is not one of Ogg’s primes it cannot be a divisor of #M. However, the other implication remains mysterious.

Duncan and Ono say:

“This discussion does not prove that every pOgg divides #M. It merely explains how the first principles of moonshine suggest this implication. Monstrous moonshine is the proof. Does this then provide a completely satisfactory solution to Ogg’s problem? Maybe or maybe not. Perhaps someone will one day furnish a map from the characteristic p supersingular j-invariants to elements of order p where the group structure of M is apparent.”

I don’t know whether they claimed the bottle, anyway.

But then, what is the non-commutative Jack Daniels Problem?

A footnote on the first page of Conway and Norton’s ‘Monstrous Moonshine’ paper says:

“Very recently, A. Pizer has shown these primes are the only ones that satisfy a certain conjecture of Hecke from 1936 relating modular forms of weight 2 to quaternion algebra theta-series.”

Pizer’s paper is “A note on a conjecture of Hecke”.

Maybe there’s a connection between monstrous moonshine and the arithmetic of integral quaternion algebras. Some hints:

The commutation relations in the Big Picture are reminiscent of the meta-commutation relations for Hurwitz quaternions, originally due to Conway in his booklet on Quaternions and Octonions.

The fact that the p-tree in the Big Picture has valency p+1 comes from the fact that the Brauer-Severi of M2(Fp) is PFp1. In fact, the Big Picture should be related to the Brauer-Severi scheme of M2(Z).

Then, there’s Jorge Plazas claiming that Connes-Marcolli’s GL2-system might be related to moonshine.

One of the first things I’ll do when I return is to run to the library and get our copy of Shimura’s ‘Introduction to the arithmetic theory of automorphic functions’.

Btw. the bottle in the title image is not a Jack Daniels but the remains of a bottle of Ricard, because I’m still in the French mountains.

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the 171 moonshine groups

Monstrous moonshine associates to every element of order n of the monster group M an arithmetic group of the form
(n|h)+e,f,
where h is a divisor of 24 and of n and where e,f, are divisors of nh coprime with its quotient.

In snakes, spines, and all that we’ve constructed the arithmetic group
Γ0(n|h)+e,f,
which normalizes Γ0(N) for N=h.n. If h=1 then this group is the moonshine group (n|h)+e,f,, but for h>1 the moonshine group is a specific subgroup of index h in Γ0(n|h)+e,f,.

I’m sure one can describe this subgroup explicitly in each case by analysing the action of the finite group (Γ0(n|h)+e,f,)/Γ0(N) on the (N|1)-snake. Some examples were worked out by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.

But at the moment I don’t understand the general construction given by Conway, McKay and Sebbar in On the discrete groups of moonshine. I’m stuck at the last sentence of (2) in section 3. Nothing a copy of Charles Ferenbaugh Ph. D. thesis cannot fix.

The correspondence between the conjugacy classes of the Monster and these arithmetic groups takes up 3 pages in Conway & Norton’s Monstrous Moonshine. Here’s the beginning of it.

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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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