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).