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Computing the endomorphism ring of an elliptic curve over a number field

John E. Cremona, Andrew V. Sutherland

TL;DR

This paper addresses the problem of recognizing Hilbert class polynomials and extracting CM data for elliptic curves over number fields. It introduces two new algorithms: a deterministic, GRH-assisted approach and a Las Vegas probabilistic method that computes the CM discriminant when applicable, both aiming to compute the endomorphism ring efficiently. The work situates these algorithms within a broad theoretical framework involving CM theory, Hilbert class polynomials, Galois actions, modular polynomials, and Galois representations, and demonstrates substantial practical improvements over existing tools. The methods are implemented in Magma, Pari/GP, and SageMath, with empirical results showing near-quasilinear performance on inputs where previous methods were slower by orders of magnitude, hence enabling faster endomorphism computations for elliptic curves over number fields.

Abstract

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.

Computing the endomorphism ring of an elliptic curve over a number field

TL;DR

This paper addresses the problem of recognizing Hilbert class polynomials and extracting CM data for elliptic curves over number fields. It introduces two new algorithms: a deterministic, GRH-assisted approach and a Las Vegas probabilistic method that computes the CM discriminant when applicable, both aiming to compute the endomorphism ring efficiently. The work situates these algorithms within a broad theoretical framework involving CM theory, Hilbert class polynomials, Galois actions, modular polynomials, and Galois representations, and demonstrates substantial practical improvements over existing tools. The methods are implemented in Magma, Pari/GP, and SageMath, with empirical results showing near-quasilinear performance on inputs where previous methods were slower by orders of magnitude, hence enabling faster endomorphism computations for elliptic curves over number fields.

Abstract

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.
Paper Structure (20 sections, 20 theorems, 23 equations, 1 table, 2 algorithms)

This paper contains 20 sections, 20 theorems, 23 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Elliptic curves and their j-invariants
  4. Basic CM facts
  5. Hilbert Class Polynomials and their properties
  6. Galois theoretic constraints
  7. Galois representation constraints
  8. Modular polynomials
  9. Explicit bounds
  10. Existing Algorithms
  11. Magma's algorithm
  12. Sage's algorithms
  13. Charles' Algorithms
  14. Direct approach
  15. Randomized supersingularity testing
  16. ...and 5 more sections

Key Result

Theorem 1

Given a monic irreducible polynomial $H\in \mathbb{Z}[X]$, if the GRH is true then Algorithm alg:one returns true if and only if $H$ is a Hilbert class polynomial. It admits a deterministic implementation that runs in $(h^2|H|)^{1+o(1)}$ time using $(h|H|)^{1+o(1)}$ space.

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Proposition 4
  • proof
  • Remark 5
  • Proposition 6
  • Remark 7
  • Remark 8
  • Proposition 9
  • ...and 41 more