Quadratic polynomials P(F(q),F(q^{2}))=0
for Hauptmoduln F of level 4N
Suppose G is a genuszero congruence group
commensurable with PSL_{2}(Z)
that is 2adically contained in Gamma_{0}(2).
Let F be a Hauptmodul for the associated modular curve.
Then F(q) and F(q^{2})
satisfy a polynomial equation
P(F(q),F(q^{2}))=0
quadratic in both variables, and
P(X_{i} , X_{i+1})=0
gives a recursive tower of modular curves. We exhibited
fifteen such examples, where G is 2adically Gamma_{0}(2),
there.
Here we give the six cases where G contains some group
Gamma_{0}(4N) but is 2adically
Gamma_{0}(2^{r}) for some r>1.
In each case we choose a Hauptmodul that vanishes at a finite cusp
and is q^{1} + O(1) at infinity,
and express it as a product of functions
(eta(q^{n}))^{bn}.
The largest r that occurs is 4.
Here G=Gamma_{0}(16), of index 24 in Gamma(1).
We can use the Hauptmodul with
b_{4}=2, b_{8}=6,
and b_{16}=4, and all other b_{n}=0.
Then
P(A,B) =
B^{2}  A^{2}B + 4.
equivalently, subtracting 2 to put the zero of the Hauptmodul
at the zero cusp, we have
b_{1}=2, b_{2}=1,
b_{8}=1, b_{16}=2
(and other b_{n}=0), and quadratic relation
P(A,B) =
B^{2}  A(A+4)(B+2).
Likewise r=3 allows only G=Gamma_{0}(8),
of index 12. The Hauptmodul with b_{2}=4,
b_{4}=12, and b_{8}=8,
and all other b_{n}=0, yields
P(A,B) =
(B+4)^{2}  A^{2}B.
[This Hauptmodul is equivalent to the square of our first
Gamma_{0}(16) Hauptmodul under a conjugation in
PSL_{2}(Z).] Equivalently,
subtracting 4 to put the zero of the Hauptmodul
at the zero cusp, we have
b_{1}=4, b_{2}=2,
b_{4}=2, b_{8}=4
(and other b_{n}=0), and quadratic relation
P(A,B) =
B^{2}  A(A+8)(B+4).
This is also a guise of the classical recursion for the AGM
(arithmeticgeometric mean), as may be seen by the further substitution
(A,B)=(8/(X1),8/(Y1)), under which the equation becomes
4XY^{2} = (X+1)^{2},
that is Y = (1+X) / (2 sqrt(X)).
In the case r=2 we find four possibilities,
which we describe as we did for the fifteen cases of
r=1.
Define
V_{n}(q) :=
eta(q^{m}) / eta(q^{4m}).
For each of the four groups, we write a Hauptmodul
as a finite product of functions
(V_{n}(q))^{an} :
G 
index 
nonzero a_{n} 
equation 
Gamma_{0}(4) 
6 
a_{1} = 8 
B^{2} =
A (A + 16) (B + 16)

Gamma_{0}(12) : w_{3} 
12 
a_{1} = a_{3} = 2 
B^{2} =
A (A + 4) (B + 4)

Gamma_{0}(28) : w_{7} 
24 
a_{1} = a_{7} = 1 
B^{2} =
A (A + 2) (B + 2)

Gamma_{0}(12) 
24 
a_{1} = 1, a_{3} = 3 
B^{2} =
A (A  2) (B  2)

As with
r=1,
there are G that are commensurable with,
but not subgroups of, Gamma(1); the ``index'' of such a G
is the quotient of the indices in Gamma(1) and G
of a common subgroup of finite index.
Note that both Gamma_{0}(12):w_{3}
and its index2 subgroup Gamma_{0}(12) arise.
Thus w_{3} yields an involution of the
Gamma_{0}(12) tower, acting in the same way on each coordinate
X_{i}; if we apply to this involution to one
of the variables A,B we get an alternative recursion
giving the same tower. We calculate that this involution
takes X to (X4)/(X1).
Applying this involution to one of A,B
and adding 1 to both variables, we find the equivalent recursion
A^{2}(B^{2}+3B)=9(B1).