[Paper Review] Homogeneous division polynomials for Weierstrass elliptic curves

Jinbi Jin|arXiv (Cornell University)|Mar 18, 2013

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper constructs homogeneous division polynomials αₙ, βₙ, γₙ in ℤ[a₁,a₂,a₃,a₄,a₆][x,y,z] that define multiplication by n on all Weierstrass elliptic curves over arbitrary rings, using the Theorem of the Cube to prove their existence and universality. The key contribution is showing that equations for the modular curve Y₁(n) over ℂ, such as those by Baaziz, in fact define Y₁(n) over ℤ[1/n].

ABSTRACT

Starting from the classical division polynomials we construct homogeneous polynomials $α_n$, $β_n$, $γ_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP = \bigl(α_n(P):β_n(P):γ_n(P)\bigr)$. To show that $α_n,β_n,γ_n$ indeed have this property we use the a priori existence of such polynomials, which we deduce from the Theorem of the Cube. We then use this result to show that the equations defining the modular curve $Y_1(n)_{\mathbb C}$ computed for example by Baaziz, in fact are equations of $Y_1(n)$ over $\mathbb Z[1/n]$.

Motivation & Objective

• To construct homogeneous polynomials αₙ, βₙ, γₙ that define multiplication by n on all Weierstrass elliptic curves over arbitrary rings, including non-affine points.
• To resolve the limitation of classical division polynomials, which are only defined on affine points (z=1), by providing a homogeneous, projective formulation.
• To prove that the classical equations for the modular curve Y₁(n) over ℂ, such as those by Baaziz, in fact define Y₁(n) over ℤ[1/n], not just over ℂ.
• To establish that the universal smooth Weierstrass curve over ℤ[1/n] admits a canonical model with a universal n-torsion point via these homogeneous polynomials.
• To show that the homogeneous polynomials αₙ, βₙ, γₙ are uniquely determined up to sign and have degree n², generalizing the classical affine case.

Proposed method

• Constructs αₙ, βₙ, γₙ as homogenizations of the classical division polynomials ΦₙΨₙ, Ωₙ, Ψₙ³ modulo the Weierstrass equation, ensuring degree n² and homogeneity.
• Uses the Theorem of the Cube to prove the a priori existence of such homogeneous polynomials defining multiplication by n on smooth Weierstrass curves over arbitrary rings.
• Reduces the problem to the universal smooth Weierstrass curve and uses the Theorem of the Cube to show that the polynomials αₙ, βₙ, γₙ define multiplication by n on the universal curve.
• Extends the result from smooth curves to all Weierstrass curves (including singular ones) by showing the polynomials are well-defined and compatible under base change.
• Applies the construction to the moduli problem of elliptic curves with a Z/nZ-embedding, using the Tate normal form to show representability over ℤ[1/n].
• Defines fₙ ∈ ℤ[1/n][s,t] recursively via ψₙ and previous fₙ, showing that fₙ = 0 defines Y₁(n) over ℤ[1/n] when combined with δ ≠ 0 (discriminant condition).

Experimental results

Research questions

• RQ1 Can homogeneous polynomials αₙ, βₙ, γₙ be constructed over ℤ[a₁,a₂,a₃,a₄,a₆][x,y,z] such that nP = (αₙ(P) : βₙ(P) : γₙ(P)) for all Weierstrass curves over arbitrary rings and all points P = (x:y:z)?
• RQ2 Does the use of the Theorem of the Cube allow for a conceptual proof of the existence and universality of such homogeneous division polynomials?
• RQ3 Do the classical equations for the modular curve Y₁(n) over ℂ, such as those by Baaziz, in fact define Y₁(n) over ℤ[1/n], rather than just over ℂ?
• RQ4 Is the moduli space of elliptic curves with a Z/nZ-embedding representable over ℤ[1/n] via the vanishing of fₙ and non-vanishing discriminant?
• RQ5 Can the Tate normal form be used to show that the universal n-torsion point corresponds to (0:0:1) on a Weierstrass curve defined over ℤ[1/n][s,t,δ⁻¹]?

Key findings

• The homogeneous polynomials αₙ, βₙ, γₙ exist and are uniquely determined up to sign, with each of degree n², and satisfy nP = (αₙ(P) : βₙ(P) : γₙ(P)) for all Weierstrass curves over any ring and all points P = (x:y:z).
• The Theorem of the Cube provides a conceptual proof of the existence of such polynomials, ensuring they define multiplication by n on the universal smooth Weierstrass curve.
• The equations fₙ = 0 and δ ≠ 0, where fₙ is defined recursively from ψₙ = Ψₙ(1+s,t,t,0,0,0,0), define the modular curve Y₁(n) over ℤ[1/n], not just over ℂ.
• The moduli problem P(n) of elliptic curves with a Z/nZ-embedding is representable over ℤ[1/n] by Spec ℤ[1/n][s,t,δ⁻¹,1/n]/(fₙ), with the universal curve given by y²z + (1+s)xyz + t yz² = x³ + t x² z.
• The universal n-torsion point corresponds to (0:0:1) on this model, and the construction confirms that fₙ vanishes precisely at n-torsion points over ℤ[1/n].
• The construction shows that the classical equations for Y₁(n) over ℂ, such as those in Baaziz (2010), are valid over ℤ[1/n], extending their arithmetic significance.