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Using Fricke modular polynomials to compute isogenies

François Morain

Abstract

Let $\mathcal{E}$ be an elliptic curve over a field $\mathbf{K}$ and $\ell$ a prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$ by an isogeny of degree $\ell$ only if $Φ_\ell^t(X, j(\mathcal{E})) = 0$, where $Φ_\ell^t(X, Y)$ is the traditional modular polynomial. Moreover, $Φ_\ell^t$ gives the coefficients of $\mathcal{E}^*$, together with parameters needed to build the isogeny explicitly. Since $Φ_\ell^t$ has very large coefficients, many families with smaller coefficients can be used instead, as described by Elkies, Atkin and others. In this work, we concentrate on the computation of the family of modular polynomials introduced by Fricke and more recently used by Charlap, Coley and Robbins. In some cases, the resulting polynomials are small, which justifies the interest of this study. We review and adapt the known algorithms to perform the computations of these polynomials. After describing the use of series computations, we investigate fast algorithms using floating point numbers based on fast numerical evaluation of Eisenstein series. We also explain how to use isogeny volcanoes as an alternative. The last part is concerned with finding explicit formulas for computing the coefficients of $\mathcal{E}^*$. To this we add tables of numerical examples.

Using Fricke modular polynomials to compute isogenies

Abstract

Let be an elliptic curve over a field and a prime. There exists an elliptic curve related to by an isogeny of degree only if , where is the traditional modular polynomial. Moreover, gives the coefficients of , together with parameters needed to build the isogeny explicitly. Since has very large coefficients, many families with smaller coefficients can be used instead, as described by Elkies, Atkin and others. In this work, we concentrate on the computation of the family of modular polynomials introduced by Fricke and more recently used by Charlap, Coley and Robbins. In some cases, the resulting polynomials are small, which justifies the interest of this study. We review and adapt the known algorithms to perform the computations of these polynomials. After describing the use of series computations, we investigate fast algorithms using floating point numbers based on fast numerical evaluation of Eisenstein series. We also explain how to use isogeny volcanoes as an alternative. The last part is concerned with finding explicit formulas for computing the coefficients of . To this we add tables of numerical examples.
Paper Structure (47 sections, 34 theorems, 160 equations, 2 figures, 1 table, 11 algorithms)

This paper contains 47 sections, 34 theorems, 160 equations, 2 figures, 1 table, 11 algorithms.

Table of Contents

  1. Introduction
  2. Modular forms
  3. Elements of theory
  4. Integer matrices
  5. Modular forms
  6. Eisenstein series
  7. Formulas
  8. Expressing modular forms as polynomials
  9. Modular polynomials from modular forms
  10. General results
  11. Orders and heights
  12. The prime case
  13. The case of $F_\ell$
  14. Fast numerical evaluation of Eisenstein series
  15. Jacobi $\theta$ functions
  16. ...and 32 more sections

Key Result

Proposition 2.1

The matrices with $a_i > 0$, $a_i d_i = n$, $\gcd(a_i, b_i, d_i)=1$ and $0 \leq b_i < d_i$, form a set of representatives of $\mathcal{M}_n$ modulo $\sim_\Gamma$.

Figures (2)

  • Figure 1: Examples of grids and sets $\mathcal{I}_\epsilon$.
  • Figure 2: A typical set of volcanoes.

Theorems & Definitions (34)

  • Proposition 2.1: Prop. 6.5.3
  • Proposition 2.2: Prop. 6.5.3 -- cont'd
  • Proposition 2.3: Ex. 6.2.12
  • Lemma 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Proposition 2.9
  • Corollary 2.10
  • ...and 24 more