[Paper Review] Complex Multiplication Tests for Elliptic Curves

Denis Charles|ArXiv.org|Sep 26, 2004

Cryptography and Residue Arithmetic14 references3 citations

TL;DR

This paper presents a deterministic polynomial-time algorithm to test whether an elliptic curve over a number field has complex multiplication (CM), leveraging the solvability of the Galois image of ℓ-adic representations. It proves that for sufficiently large primes ℓ, the Galois group of the ℓ-torsion field is solvable if and only if the curve has CM, enabling a decision procedure with effective bounds via Masser-W"ustholz theorems, though the multiplicative constant in the running time remains non-effective.

ABSTRACT

We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.

Motivation & Objective

• To develop an efficient algorithm to determine whether an elliptic curve over a number field has complex multiplication (CM).
• To provide a rigorous, non-heuristic alternative to prior probabilistic methods for CM testing.
• To establish a deterministic polynomial-time algorithm based on the image of ℓ-adic Galois representations.
• To analyze the conditions under which the Galois image of the ℓ-torsion field becomes solvable, linking this to the CM property.

Proposed method

• Use the Galois representation ρℓ associated with the ℓ-torsion points of the elliptic curve to analyze the structure of Gal(L(E[ℓ])/L).
• Leverage the fact that for ℓ ≥ 5, GL₂(𝔽ℓ) is non-solvable, while the image of ρℓ is solvable if and only if the curve has CM.
• Apply the Masser-Wüstholz theorem to bound the minimal prime ℓ for which ρℓ(G_L) contains SL₂(𝔽ℓ) when E does not have CM.
• Check solvability of the extension L(E[ℓ])/L using computational algebra tools, such as the algorithm of Landau and Miller, for a fixed ℓ.
• Use the Weil height of the j-invariant to bound the input size and ensure polynomial-time complexity.
• Select ℓ > c(max{d, h})^γ for effective constants c, γ to guarantee that non-CM curves yield non-solvable Galois images.

Experimental results

Research questions

• RQ1 Can the complex multiplication property of an elliptic curve over a number field be tested in deterministic polynomial time?
• RQ2 What is the minimal prime ℓ such that the Galois image of the ℓ-torsion representation becomes non-solvable for non-CM curves?
• RQ3 How can the solvability of the Galois group Gal(L(E[ℓ])/L) be used as a criterion for detecting CM?
• RQ4 To what extent can the Weil height of the j-invariant be used to bound the running time of CM testing algorithms?
• RQ5 Can the randomized algorithm based on supersingular primes be converted into a one-sided error algorithm using the results on Galois representations?

Key findings

• The paper establishes a deterministic polynomial-time algorithm to test complex multiplication in elliptic curves over number fields.
• For ℓ > c(max{d, h})^γ with effective constants c, γ, the Galois group Gal(L(E[ℓ])/L) is solvable if and only if the curve has CM.
• The algorithm relies on the fact that the image of the ℓ-adic Galois representation is solvable precisely when the curve has CM, and non-solvable otherwise for large ℓ.
• The Weil height of the j-invariant is bounded polynomially by the input size, ensuring polynomial-time encoding.
• The method is rigorous and does not rely on unproven heuristic assumptions, unlike prior approaches.
• Although the running time is polynomial, the multiplicative constant in the exponent remains non-effective due to the current state of bounds on class numbers.