Mobius aMUsements

[John Conway]

On Fri, 4 May 2001, Marc LeBrun wrote:

    [...]

>    A1615:    1,3,4,6,6,12,8,12,12,18,12,24,14,24,24,24,18,36,20,36,32,36,...
>
> This sequence is described in the EIS as "Sublattices of index n in generic
> 2-dimensional lattice; also index of GAMMA_0(n) in SL(2,z)."  (Anyone care
> to roughly decipher this definition?)

    Well, first let me say that since it's given by the formula

        N  times the product of  1 + 1/p  (over primes  p|N )

it's often called  phi+(N).

    The 2-dimensional lattice  <V,W>  consists of all mectors of the form
mV + nW,  (m,n  integers).  It has precisely  phi+(2) = 3  sublattices of
index 2,  namely  <2V,W>, <V,2W> and  <V+W,2V> (which = <V+W,2W>),  and
so for other indices.

    SL2(Z) = Gamma  is the group of all 2x2 matrices

         / a  b \
        (        )    where  a,b,c,d  are integers  with  ad-bc = 1,
         \ c  d /

and Gamma0(N) is usually defined as the subgroup of this for which  N|c.

   But conceptually  Gamma is best thought of as the group of (positive)
automorphisms of a lattice  <V,W>,  its typical element taking
V -> aV + bW,  W -> cV + dW,  and then  Gamma0(N)  can be defined as
the subgroup consisting of the automorphisms that fix the sublattice
<NV,W>  ( or should that be <V,NW> ? ) of index N.

     Regards,  John Conway