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Computing Isogenies at Singular Points of the Modular Polynomial

William E. Mahaney, Travis Morrison

TL;DR

The isogeny $\phi$ can then be efficiently computed from $E$ and $\widetilde{E}$ using an algorithm of Bostan, Morain, Salvy, and Schost.

Abstract

In this paper we present a method which, given a singular point $(j_1, j_2)$ on $Y_0(\ell)$ with $j_1, j_2 \neq 0, 1728$ and an elliptic curve $E$ with $j$-invariant ${j_1}$, returns an elliptic curve $\widetilde{E}$ with $j$-invariant ${j_2}$ that admits a normalized $\ell$-isogeny $φ\colon E\to \widetilde{E}$. The isogeny $φ$ can then be efficiently computed from $E$ and $\widetilde{E}$ using an algorithm of Bostan, Morain, Salvy, and Schost.

Computing Isogenies at Singular Points of the Modular Polynomial

TL;DR

The isogeny can then be efficiently computed from and using an algorithm of Bostan, Morain, Salvy, and Schost.

Abstract

In this paper we present a method which, given a singular point on with and an elliptic curve with -invariant , returns an elliptic curve with -invariant that admits a normalized -isogeny . The isogeny can then be efficiently computed from and using an algorithm of Bostan, Morain, Salvy, and Schost.
Paper Structure (10 sections, 5 theorems, 43 equations, 1 algorithm)

This paper contains 10 sections, 5 theorems, 43 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Elliptic Curves
  4. Modular polynomials, curves, and functions
  5. Singularities of Plane Curves
  6. Elkies' method to compute the isogenous curve
  7. Singular points on Y0(l)
  8. Computing Reductions of j '
  9. Pseudocode and examples
  10. Acknowledgements

Key Result

Proposition 4.1

Suppose $j_1,j_2$ are two elements of $k$, neither of which are $0$ or $1728$. Let $(j_1,j_2)$ be a point on $Y_0(\ell)$ and let $m$ be the number of $\ell$-isogenies with distinct kernels between two fixed elliptic curves $E_1$ and $E_2$ with $j$-invariants $j_1$ and $j_2$. If $\mathop{\mathrm{\ope

Theorems & Definitions (16)

  • Definition 2.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Definition 5.1
  • Remark 5.2
  • Proposition 5.3
  • proof
  • Example 5.4
  • Proposition 5.5
  • ...and 6 more