Explicit constructions of cyclic N-isogenies

Abstract

The modular curve X_0(N) parametrizes elliptic curves together with a cyclic subgroup of order N, and hence cyclic N-isogenies. While explicit moduli descriptions of X_1(N) are well developed, a comparable construction for X_0(N) has remained incomplete. We give a uniform method for constructing explicit generators of C(X_0(N)), extending an approach of Dowd, and use them to obtain a concrete moduli interpretation of cyclic N-isogenies. This yields explicit formulas for sporadic rational points on X_0(N) and the associated isogenies, providing a unified solution to the moduli problem for X_0(N).