## Quadratic polynomials P(F(q),F(q2))=0 for Hauptmoduln F of level 4N

Suppose G is a genus-zero congruence group commensurable with PSL2(Z) that is 2-adically contained in Gamma0(2). Let F be a Hauptmodul for the associated modular curve. Then F(q) and F(q2) satisfy a polynomial equation P(F(q),F(q2))=0 quadratic in both variables, and P(Xi , Xi+1)=0 gives a recursive tower of modular curves. We exhibited fifteen such examples, where G is 2-adically Gamma0(2), there. Here we give the six cases where G contains some group Gamma0(4N) but is 2-adically Gamma0(2r) 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(qn))bn.

The largest r that occurs is 4. Here G=Gamma0(16), of index 24 in Gamma(1). We can use the Hauptmodul with b4=-2, b8=6, and b16=-4, and all other bn=0. Then

P(A,B) = B2 - A2B + 4.
equivalently, subtracting 2 to put the zero of the Hauptmodul at the zero cusp, we have b1=2, b2=-1, b8=1, b16=-2 (and other bn=0), and quadratic relation
P(A,B) = B2 - A(A+4)(B+2).

Likewise r=3 allows only G=Gamma0(8), of index 12. The Hauptmodul with b2=-4, b4=12, and b8=-8, and all other bn=0, yields

P(A,B) = (B+4)2 - A2B.
[This Hauptmodul is equivalent to the square of our first Gamma0(16) Hauptmodul under a conjugation in PSL2(Z).] Equivalently, subtracting 4 to put the zero of the Hauptmodul at the zero cusp, we have b1=4, b2=-2, b4=2, b8=-4 (and other bn=0), and quadratic relation
P(A,B) = B2 - A(A+8)(B+4).
This is also a guise of the classical recursion for the AGM (arithmetic-geometric mean), as may be seen by the further substitution (A,B)=(8/(X-1),8/(Y-1)), under which the equation becomes
4XY2 = (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

Vn(q) := eta(qm) / eta(q4m).
For each of the four groups, we write a Hauptmodul as a finite product of functions (Vn(q))an :
 G index nonzero an equation Gamma0(4) 6 a1 = 8 B2 = A (A + 16) (B + 16) Gamma0(12) : w3 12 a1 = a3 = 2 B2 = A (A + 4) (B + 4) Gamma0(28) : w7 24 a1 = a7 = 1 B2 = A (A + 2) (B + 2) Gamma0(12) 24 a1 = -1,   a3 = 3 B2 = 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 Gamma0(12):w3 and its index-2 subgroup Gamma0(12) arise. Thus w3 yields an involution of the Gamma0(12) tower, acting in the same way on each coordinate Xi; 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 (X-4)/(X-1). Applying this involution to one of A,B and adding 1 to both variables, we find the equivalent recursion A2(B2+3B)=9(B-1).