## 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) : 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.