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Roots of unity and the Big Picture

All lattices in the moonshine picture are number-like, that is of the form Mgh with M a positive integer and 0g<h with (g,h)=1. To understand the action of the Bost-Connes algebra on the Big Picture it is sometimes better to view the lattice Mgh as a primitive h-th root of unity, centered at hM.

The distance from M to any of the lattices Mgh is equal to 2log(h), and the distances from M and Mgh to hM are all equal to log(h).

For a prime value h, these h lattices are among the h+1 lattices branching off at hM in the h-adic tree (the remaining one being h2M).

For general h the situation is more complex. Here’s the picture for h=6 with edges in the 2-adic tree painted blue, those in the 3-adic tree red.

\xymatrix{& & M \frac{1}{2} \ar@[blue]@{-}[d] & \\ 
& M \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & 2M \ar@[red]@{-}[d] & M \frac{1}{6} \ar@[red]@{-}[d] \\ 
M \frac{1}{3} \ar@[red]@{-}[r] & 3M \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & \boxed{6 M} \ar@[blue]@{-}[r] & 3M \frac{1}{2} \ar@[red]@{-}[d]  \\ & M \frac{2}{3} & & M \frac{5}{6}}

To describe the moonshine group (n|h)+e,f, (an example was worked out in the tetrahedral snake post), we need to study the action of base-change with the matrix
x=[11h01]
which sends a lattice of the form Mgh with 0g<h to Mg+Mh, so is a rotation over 2πMh around hM. But, we also have to describe the base-change action with the matrix y=[10n1] and for this we better use the second description of the lattice as Mgh=(gh,1h2M) with g the multiplicative inverse of g modulo h. Under the action by y, the second factor 1h2M will be fixed, so this time we have to look at all lattices of the form (gh,1h2M) with 0g<h, which again can be considered as another set of h-th roots of unity, centered at hM. Here's this second interpretation for h=6: \xymatrix{M \frac{5}{6} \ar@[red]@{-}[d] & & 4M \frac{1}{3} \ar@[red]@{-}[d] & \\
3M \frac{1}{2} \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & \boxed{6M} \ar@[blue]@{-}[r] \ar@[red]@{-}[d] & 12 M \ar@[red]@{-}[r] \ar@[red]@{-}[d] & 4 M \frac{2}{3} \\
M \frac{1}{6} & 18 M \ar@[blue]@{-}[r] \ar@[blue]@{-}[d] & 36 M & \\
& 9M \frac{1}{2} & & }
Under x the first set of h-th roots of unity centered at hM is permuted, whereas y permutes the second set of h-th roots of unity.
These interpretations can be used to spot errors in computing the finite groups Γ0(n|h)/Γ0(n.h).

Here’s part of the calculation of the action of y on the (360|1)-snake (which consists of 60-lattices).

First I got a group of order roughly 600.000. After correcting some erroneous cycles, the order went down to 6912.

Finally I spotted that I mis-numbered two lattices in the description of x and y, and the order went down to 48 as it should, because I knew it had to be equal to C2×C2×A4.

Published in groups noncommutative

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