Quadratic polynomials P(F(q),F(q2))=0
for Hauptmoduln F of singly even level
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 exhibit fifteen such examples.
Let
u(q) := eta(q)/eta(q2)
= q-1/24 /
(1+q)(1+q2)(1+q3)(1+q4)...,
Um := u(qm).
So for instance
H=(U1(q))24
is a Hauptmodul for X0(2).
We know that (A,B) =
(H(q),H(q2))
satisfy the equation
B2 = 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(qn))an,
with all n's odd.
[Note that necessarily sum(n an)=24 and
(since every n is odd) G is 2-adically Gamma(1).
The odd condition also means that the sum a of the
an has the same parity as the sum of
n an, and is thus even.]
Then
H(q)
:= f(q) / f(q2)
= product of (Un(q))an
is a Hauptmodul for the intersection G
of G' with Gamma0(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(q2) is
B2 = A
(AB + 2a1 B + c),
where c is either 2a/2
or -2a/2.
Here the coefficient 2a1 is obtained by comparing
q-expansions, and |c| is determined
by using an involution w2,
which switches H with c/H.
(It turns out that the sign of c always coincides
with the sign of the first nonzero an.)
The curve
P(Xi , Xi+1)=0
with m variables then corresponds to the intersection
of G' with Gamma0(2m+1).
We found fifteen examples of such f. For each one,
we list the coefficients 2a1 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 one-third the index of G),
and all nonzero an.
| 2a1 |
c |
G' |
index |
nonzero an |
| 48 |
4096 |
Gamma(1) |
1 |
a1 = 24 |
| 12 |
64 |
Gamma0(3) : w3 |
2 |
a1 = a3 = 6 |
| 8 |
16 |
Gamma0(5) : w5 |
3 |
a1 = a5 = 4 |
| 6 |
8 |
Gamma0(7) : w7 |
4 |
a1 = a7 = 3 |
| 4 |
4 |
Gamma0(11) : w11 |
6 |
a1 = a11 = 2 |
| 2 |
2 |
Gamma0(23) : w23 |
12 |
a1 = a23 = 1 |
| -6 |
-8 |
Gamma0(3) |
4 |
a1 = -3,
a3 = 9 |
| -2 |
-4 |
Gamma0(5) |
6 |
a1 = -1,
a5 = 5 |
| 6 |
4 |
Gamma0(9) : w9 |
6 |
a1 = a9 = 3,
a3 = -2 |
| 2 |
4 |
Gamma0(15) :
<w3 , w5> |
6 |
a1 = a3 =
a5 = a15 = 1
|
| 4 |
2 |
Gamma0(15) : w15 |
12 |
a1 = a15 = 2,
a3 = a5 = -1 |
| 0 |
16 |
[9] |
3 |
a3 = 8 |
| 0 |
4 |
[27] |
6 |
a3 = a9 = 2 |
| 0 |
2 |
[63] |
12 |
a3 = a21 = 1 |
| 0 |
-2 |
Gamma0(9) |
12 |
a3 = -1, a9 = 3 |
The groups denoted [9], [27], [63]
are conjugate in PSL2(Q)
with the intersection of Gamma(1),
Gamma0(3):w3,
Gamma0(7):w7
with the j1/3 group.
[This last is also the index-3 congruence group
corresponding to the normal subgroup of index 3 in
PSL2(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 A3 and B3 where
A,B satisfy the quadratic relation for
Gamma(1), Gamma0(3):w3,
Gamma0(7):w7 respectively.
The last group is likewise related
with Gamma0(3), but here the intersection
with the j1/3 group is just Gamma(3),
which the same PSL2(Q)-conjugation
transforms into the familiar Gamma0(9).
There are four cases in the table of a group G'
contained with index 2 in a larger tabulated group
G':wl. In that case,
wl yields an involution of the G' 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.
These cases are:
G' = Gamma0(3),
l = 3,
w3(X) = (8-X)/(1+X) ;
G' = Gamma0(5),
l = 5,
w3(X) = (4-X)/(1+X) ;
G' = Gamma0(9),
l = 9,
w3(X) = (2-X)/(1+X) ;
G' = Gamma0(15) : w15,
l = 3 or 5,
w3(X) =
w5(X) =
(-2-X)/(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.