Dedekind’s Psi-function
is the index of the congruence subgroup in the modular group , is the number of points in the projective line , is the number of classes of -dimensional lattices at hyperdistance in Conway’s big picture from the standard lattice , is the number of admissible maximal commuting sets of operators in the Pauli group of a single qudit.
The first and third interpretation have obvious connections with Monstrous Moonshine.
Conway’s big picture originated from the desire to better understand the Moonshine groups, and Ogg’s Jack Daniels problem
asks for a conceptual interpretation of the fact that the prime numbers such that
Here’s a nice talk by Ken Ono : Can’t you just feel the Moonshine?
For this reason it might be worthwhile to make the connection between these two concepts and the number of points of
Surely all of this is classical, but it is nicely summarised in the paper by Tatitscheff, He and McKay “Cusps, congruence groups and monstrous dessins”.
The ‘monstrous dessins’ from their title refers to the fact that the lattices
Here’s the ‘monstrous dessin’ for

But, one can compute these dessins for arbitrary
We will get there eventually, but let’s start at an easy pace and try to describe the points of the projective line
Over a field

Over an arbitrary (commutative) ring
with respect to scalar multiplication by units in
For
The problem is to find a canonical representative in each class in an efficient way because this is used a huge number of times in working with modular symbols.
Perhaps the best algorithm, for large
For small
- Consider the action of
on and let be the set of the smallest elements in each orbit, - For each
compute the stabilizer subgroup for this action and let be the set of smallest elements in each -orbit on the set of all elements in coprime with , - Then
.
Let’s work this out for
,- The orbits on
are
and , , , , , and , is the only number coprime with , giving us , are all coprime with , and we have trivial stabilizer, giving us the points , are coprime with and under the action of they split into the orbits
giving us the points and , are coprime with , the action of gives us the orbits
and additional points and , are coprime with and under the action of we get orbits
and points and ,- Finally,
are the only coprimes with and they form a single orbit under giving us just one additional point .
This gives us all
One way to see that
and for a prime power
which shows that
Next time, we’ll connect