Explicit constructions of cyclic N-isogenies

TL;DR

This paper develops a uniform method to explicitly construct generators for the modular curve X_0(N), providing concrete formulas for cyclic N-isogenies and rational points, thus completing the moduli description for X_0(N).

Contribution

It introduces a new uniform approach to explicitly describe X_0(N) and cyclic N-isogenies, extending prior methods and offering explicit formulas and rational points.

Findings

Explicit generators for C(X_0(N)) are constructed.

Concrete formulas for sporadic rational points are provided.

A unified solution to the moduli problem for X_0(N) is achieved.

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).