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Explicit constructions of cyclic N-isogenies

Daeyeol Jeon, Yongjae Kwon

TL;DR

The paper tackles the explicit moduli problem for $X_0(N)$ by proving that the invariants $a_4$ and $a_6$ generate the function field $\mathbb{C}(X_0(N))$ and by constructing a uniform method to recover domain and codomain elliptic curves from arbitrary points on $X_0(N)$. It extends Dowd's genus-zero approach to all levels, providing an explicit algorithm to obtain cyclic $N$-isogenies from points on $X_0(N)$ and thereby offering a complete moduli interpretation across genera. The authors apply the framework to sporadic rational points at levels including $N=11,14,15,17,19,21,27,37,43,67,163$, using genus-appropriate models and, for the challenging $N=163$, Heegner points to reconstruct the associated isogenies. This work delivers a unified, practical method for explicit cyclic $N$-isogenies with broad computational implications in the arithmetic of elliptic curves.

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).

Explicit constructions of cyclic N-isogenies

TL;DR

The paper tackles the explicit moduli problem for by proving that the invariants and generate the function field and by constructing a uniform method to recover domain and codomain elliptic curves from arbitrary points on . It extends Dowd's genus-zero approach to all levels, providing an explicit algorithm to obtain cyclic -isogenies from points on and thereby offering a complete moduli interpretation across genera. The authors apply the framework to sporadic rational points at levels including , using genus-appropriate models and, for the challenging , Heegner points to reconstruct the associated isogenies. This work delivers a unified, practical method for explicit cyclic -isogenies with broad computational implications in the arithmetic of elliptic curves.

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).
Paper Structure (8 sections, 3 theorems, 42 equations, 2 tables, 1 algorithm)

This paper contains 8 sections, 3 theorems, 42 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular curves and their moduli problems
  4. Eisenstein series and defining equations
  5. Genus $0$ case and rational construction of cyclic $N$–isogenies
  6. Constructing cyclic $N$-isogenies
  7. Generators of the function field
  8. Sporadic Cases

Key Result

Theorem 2.3

There are natural bijections Thus $Y_{0}(N)$ parameterizes elliptic curves equipped with a cyclic subgroup of order $N$, while $Y_{1}(N)$ parameterizes elliptic curves equipped with a point of order $N$.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3