The monstrous moonshine picture is the finite piece of Conway’s Big Picture needed to understand the 171 moonshine groups associated to conjugacy classes of the monster.
Last time I claimed that there were exactly 7 types of local behaviour, but I missed one. The forgotten type is centered at the number lattice
Locally around it the moonshine picture looks like this
and it involves all square roots of unity (
No, I’m not hallucinating, there are indeed
In the ‘normal’ expression
(with
and
As in the tetrahedral snake post, it is best to view the four
In the ‘normal’ expression of lattices there’s then a total of 8 different local types, but two of them consist of just one number lattice: in
Perhaps surprisingly, if we redo everything in the ‘other’ expression (and use the other families of roots of unity), then the moonshine picture has only 7 types of local behaviour. The forgotten type
I wonder what all this has to do with the action of the Bost-Connes algebra on the big picture or with Plazas’ approach to moonshine via non-commutative geometry.