Modular equations of hyperelliptic X₀(N) and an application

1. Introduction. Let N ≥ 1 be an integer and let X 0 (N) be the modular curve over Q which corresponds to the modular group Γ 0 (N). As a defining equation of X 0 (N) we have the so-called modular equation of level N. It has many good properties, e.g. it reflects the defining property of X 0 (N), it is the coarse moduli space of the isomorphism classes of the generalized elliptic curves with a cyclic subgroup of order N. But its degree and coefficients are too large to be applied to practical calculations on X 0 (N). While it is an important problem to determine the algebraic points on X 0 (N), we need a more manageable defining equation, which will also help to solve other related problems. In the case of a hyperelliptic modular curve, a kind of normal form of a defining equation is given by N. Murabayashi ([9]) and M. Shimura ([13]). In this paper, we give a relation between the modular equation of level N and the normal form in the case of a hyperelliptic modular curve X 0 (N) except for N = 40, 48. First recall that the modular equation of level N is written in the following form: where j is the modular invariant, j N (z) = j(N z), and z is the natural coordinate on H. Since X 0 (N) is hyperelliptic, it can be written in the following normal form: y 2 = f (x), f (T) ∈ Q[T ], deg f = 2g + 2, where x is a covering map of degree two from X 0 (N) to P 1 and g is the genus of X 0 (N). In this case, we obtain the following relation: j = (A(x) + B(x)y)/C(x), A(x), B(x), C(x)(= 0) ∈ Q(x). When the genus of X 0 (N) is 0, R. Fricke gave the expression for j (see [3]), and N. D. Elkies did the same when the curve X 0 (N) is elliptic or hyperelliptic where N is a prime number other than 37 ([2]). We are interested in X 0 (N) for the 19 particular values of N for which the modular curve X 0 (N) is hyperelliptic. We extend Elkies' work. We give the expression for j for 17 values of N ; to be specific, the cases N = 40, 48 are excluded. …