Monstrous Easter Egg Race

Here’s a sweet Easter egg for you to crack : a mysterious message from none other than the discoverer of Monstrous Moonshine himself…

From:  mckayj@Math.Princeton.EDU
Date:  Mon 10 Mar 2008 07:51:16 GMT+01:00
To:    lieven.lebruyn@ua.ac.be

The secret of Monstrous Moonshine and the universe. 


Let  j(q) = 1/q + 744 + sum( c[k]*q^k,k>=1) be the Fourier expansion 
at oo of the elliptic modular function.

Compute sum(c[k]^2,k=1..24) modulo 70

Background: w_25 of page x of the preface of Conway/Sloane book SPLAG 

Also in Chapter 27:
The automorphism group of the 26-dimensional Lorentzian lattice
The Weyl vector w_25 of section 2.

Jm

I realize that all of you will feel frustrated by the fact that most university libraries are closed today and possibly tomorrow, hence some help with the background material.

SPLAG of course refers to the cult-book Sphere Packings, Lattices and Groups.

26-dimensional Lorentzian space $\mathbb{R}^{25,1} $ is 26-dimensional real space equipped with the norm-map

$|| \vec{v} || = \sum_{i=1}^{25} v_i^2 – v_{26}^2 $

The Weyl vector $\vec{w}_{25} $ is the norm-zero vector in $\mathbb{R}^{25,1} $

$\vec{w}_{25} = (0,1,2,3,4,\ldots,22,23,24,70) $ (use the numerical fact that $1^2+2^2+3^2+\ldots+24^2=70^2 $)

The relevance of this special vector is that it gives a one-line description for one of the most mysterious objects around, the 24-dimensional Leech Lattice $L_{24} $. In fact

$L_{24} = \vec{w}^{\perp}/\vec{w} $ with $\vec{w}^{\perp} = { \vec{x} \in \Pi_{25,1}~:~\vec{x}.\vec{w}=0 } $

where $\Pi_{25,1} $ is the unique even unimodular lattice in $\mathbb{R}^{25,1} $. These facts amply demonstrate the moonshine nature of the numbers 24 and 70. Apart from this, the previous post may also be of use.