Quadratic polynomials P(F(q),F(q^{2}))=0
for Hauptmoduln F of singly even level
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 exhibit fifteen such examples.
Let
u(q) := eta(q)/eta(q^{2})
= q^{1/24} /
(1+q)(1+q^{2})(1+q^{3})(1+q^{4})...,
U_{m} := u(q^{m}).
So for instance
H=(U_{1}(q))^{24}
is a Hauptmodul for X_{0}(2).
We know that (A,B) =
(H(q),H(q^{2}))
satisfy the equation
B^{2} = A
(AB + 48 B + 4096).
This is the first polynomial in our family.
In general we proceed as follows. Suppose we have a congruence group
G' with a modular form f,
with a simple zero at the infinite cusp
and neither zero nor pole at any cusp not in the
G'orbit of infinity,
such that f can be written as a finite product of
(eta(q^{n}))^{an},
with all n's odd.
[Note that necessarily sum(n a_{n})=24 and
(since every n is odd) G is 2adically Gamma(1).
The odd condition also means that the sum a of the
a_{n} has the same parity as the sum of
n a_{n}, and is thus even.]
Then
H(q)
:= f(q) / f(q^{2})
= product of (U_{n}(q))^{an}
is a Hauptmodul for the intersection G
of G' with Gamma_{0}(2),
with H = q^{1} + O(1) at infinity
and with a simple zero at the cusp 0.
Moreover, the equation relating
A=H(q)
and B=H(q^{2}) is
B^{2} = A
(AB + 2a_{1} B + c),
where c is either 2^{a/2}
or 2^{a/2}.
Here the coefficient 2a_{1} is obtained by comparing
qexpansions, and c is determined
by using an involution w_{2},
which switches H with c/H.
(It turns out that the sign of c always coincides
with the sign of the first nonzero a_{n}.)
The curve
P(X_{i} , X_{i+1})=0
with m variables then corresponds to the intersection
of G' with Gamma_{0}(2^{m+1}).
We found fifteen examples of such f. For each one,
we list the coefficients 2a_{1} and c,
followed by the group G' (or its level if it does not have
a short name), its index in Gamma(1) (as a commensurable arithmetic
group  this is also onethird the index of G),
and all nonzero a_{n}.
2a_{1} 
c 
G' 
index 
nonzero a_{n} 
48 
4096 
Gamma(1) 
1 
a_{1} = 24 
12 
64 
Gamma_{0}(3) : w_{3} 
2 
a_{1} = a_{3} = 6 
8 
16 
Gamma_{0}(5) : w_{5} 
3 
a_{1} = a_{5} = 4 
6 
8 
Gamma_{0}(7) : w_{7} 
4 
a_{1} = a_{7} = 3 
4 
4 
Gamma_{0}(11) : w_{11} 
6 
a_{1} = a_{11} = 2 
2 
2 
Gamma_{0}(23) : w_{23} 
12 
a_{1} = a_{23} = 1 
6 
8 
Gamma_{0}(3) 
4 
a_{1} = 3,
a_{3} = 9 
2 
4 
Gamma_{0}(5) 
6 
a_{1} = 1,
a_{5} = 5 
6 
4 
Gamma_{0}(9) : w_{9} 
6 
a_{1} = a_{9} = 3,
a_{3} = 2 
2 
4 
Gamma_{0}(15) :
<w_{3} , w_{5}> 
6 
a_{1} = a_{3} =
a_{5} = a_{15} = 1

4 
2 
Gamma_{0}(15) : w_{15} 
12 
a_{1} = a_{15} = 2,
a_{3} = a_{5} = 1 
0 
16 
[9] 
3 
a_{3} = 8 
0 
4 
[27] 
6 
a_{3} = a_{9} = 2 
0 
2 
[63] 
12 
a_{3} = a_{21} = 1 
0 
2 
Gamma_{0}(9) 
12 
a_{3} = 1, a_{9} = 3 
The groups denoted [9], [27], [63]
are conjugate in PSL_{2}(Q)
with the intersection of Gamma(1),
Gamma_{0}(3):w_{3},
Gamma_{0}(7):w_{7}
with the j^{1/3} group.
[This last is also the index3 congruence group
corresponding to the normal subgroup of index 3 in
PSL_{2}(Z/3Z),
which in turn is isomorphic with the alternating group of order 4.]
Their quadratic polynomials can be obtained as the relations satisfied
by A^{3} and B^{3} where
A,B satisfy the quadratic relation for
Gamma(1), Gamma_{0}(3):w_{3},
Gamma_{0}(7):w_{7} respectively.
The last group is likewise related
with Gamma_{0}(3), but here the intersection
with the j^{1/3} group is just Gamma(3),
which the same PSL_{2}(Q)conjugation
transforms into the familiar Gamma_{0}(9).
There are four cases in the table of a group G'
contained with index 2 in a larger tabulated group
G':w_{l}. In that case,
w_{l} yields an involution of the G' 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.
These cases are:
G' = Gamma_{0}(3),
l = 3,
w_{3}(X) = (8X)/(1+X) ;
G' = Gamma_{0}(5),
l = 5,
w_{3}(X) = (4X)/(1+X) ;
G' = Gamma_{0}(9),
l = 9,
w_{3}(X) = (2X)/(1+X) ;
G' = Gamma_{0}(15) : w_{15},
l = 3 or 5,
w_{3}(X) =
w_{5}(X) =
(2X)/(1+X).
Note that in each case (1+X)w(1+X)
is a constant, which equals 9, 5, 3, 1 respectively in the four cases.