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It is known that in the moduli space A A of elliptic curves, there exist precisely 9 1 ‫-ޑ‬rational points corresponding to the isomorphism class of elliptic curves with complex multiplication by the ring of algebraic integers of a principal imaginary quadratic number field. Here, we prove that in the moduli space A A of principally 2 polarized abelian surfaces, there exist precisely 19 ‫-ޑ‬rational points corresponding to the isomorphism class of abelian surfaces whose endomorphism rings are isomorphic to the rings of algebraic integers of some imaginary cyclic quartic number fields. ᮊ

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:

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The purpose of this paper is to decide the conditions under which a CM elliptic curve is modular over its field of definition. r 2004 Elsevier Inc. All rights reserved.