The Wronskian of the x-coordinate of an isogeny

Let φ : W₁ → W₂ be an isogeny of elliptic curves, with x ∘ φ = r(x) for the rational function r = N / D = φ.xRatFunc in lowest terms, and suppose that φ pulls the invariant differential ω₂ = dx / u₂ back to c ω₁, where u = 2y + a₁x + a₃. Pulling ω₂ back coordinate by coordinate gives φ^*ω₂ = d(r(x)) / φ^*u₂ = r'(x) dx / φ^*u₂, and comparing with c dx / u₁ yields

w(x) u₁ = c D(x)² φ^*u₂, where w = D N' - D' N

is the Wronskian Polynomial.wronskian D N of the denominator and numerator, so that r' = w / D². In the language of Silverman III.4, when φ is separable (c ≠ 0), φ^*u₂ = s(x) u₁ for the odd part s = w / (c D²) of φ, and the identity says r' = c s. For inseparable φ both c and w vanish, and the identity carries no information about s.

Squaring it and using u² = Ψ₂Sq(x) = 4x³ + b₂x² + 2b₄x + b₆ on both curves turns it into an identity of polynomials, with no reference to y:

w² Ψ₂Sq₁ = c² D · D³ Ψ₂Sq₂(N / D),

where D³ Ψ₂Sq₂(N / D) = 4N³ + b₂'N²D + 2b₄'ND² + b₆'D³. Evaluated at a point where D does not vanish, and when c ≠ 0, this is Ψ₂Sq₂(r(x)) = s(x)² Ψ₂Sq₁(x) with s = w / (c D²). These are the two identities by which the real period integral transforms under x ↦ r(x) (WeierstrassCurve.ofReal_abs_mul_realPeriodLIntegral_eq_setLIntegral_encard).

For φ = [n] the first identity is TauCeti.Isogeny.wronskian_Φ_ΨSq_mul_invariantDifferentialDenom, there stated with the division polynomials Φₙ and ΨSqₙ in place of N and D.

Main results

• TauCeti.Isogeny.aeval_wronskian_xRatFunc_mul_invariantDifferentialDenom: w(x) u₁ = c D(x)² φ^*u₂ in the function field of W₁.
• TauCeti.Isogeny.wronskian_xRatFunc_ne_zero: for a separable isogeny (c ≠ 0) the Wronskian is nonzero.
• TauCeti.Isogeny.wronskian_xRatFunc_sq_mul_Ψ₂Sq: the polynomial identity w² Ψ₂Sq₁ = c² D (4N³ + b₂'N²D + 2b₄'ND² + b₆'D³).
• TauCeti.Isogeny.eval_wronskian_xRatFunc_sq_mul_eval_Ψ₂Sq: its value at a point x₀ with D(x₀) ≠ 0, w(x₀)² Ψ₂Sq₁(x₀) = c² D(x₀)⁴ Ψ₂Sq₂(r(x₀)).

References

• J. Silverman, The Arithmetic of Elliptic Curves, III.4 (the shape φ(x, y) = (r(x), y s(x) + t(x)) of an isogeny) and III.5 (φ^*ω₂ = c ω₁).

theorem TauCeti.Isogeny.aeval_wronskian_xRatFunc_mul_invariantDifferentialDenom {F : Type u_1} [Field F] {W₁ W₂ : WeierstrassCurve.Affine F} [WeierstrassCurve.IsElliptic W₁] [WeierstrassCurve.IsElliptic W₂] {φ : Isogeny W₁ W₂} {c : F} (hc : φ.pullbackDifferential W₂.invariantDifferential = c • W₁.invariantDifferential) : (Polynomial.aeval W₁.genericX) (φ.xRatFunc.denom.wronskian φ.xRatFunc.num) * W₁.invariantDifferentialDenom = (algebraMap F W₁.FunctionField) c * (Polynomial.aeval W₁.genericX) φ.xRatFunc.denom ^ 2 * φ.fieldPullback W₂.invariantDifferentialDenom

The Wronskian identity of an isogeny: if φ^*ω₂ = c ω₁, then

w(x) u₁ = c D(x)² φ^*u₂,

where w is the Wronskian of φ.xRatFunc = N / D and u = 2y + a₁x + a₃ is the denominator of the invariant differential.

theorem TauCeti.Isogeny.wronskian_xRatFunc_ne_zero {F : Type u_1} [Field F] {W₁ W₂ : WeierstrassCurve.Affine F} [WeierstrassCurve.IsElliptic W₁] [WeierstrassCurve.IsElliptic W₂] {φ : Isogeny W₁ W₂} {c : F} (hc : φ.pullbackDifferential W₂.invariantDifferential = c • W₁.invariantDifferential) (hc0 : c ≠ 0) : φ.xRatFunc.denom.wronskian φ.xRatFunc.num ≠ 0

The Wronskian of a separable isogeny is nonzero: if φ^*ω₂ = c ω₁ with c ≠ 0, then D N' - D' N ≠ 0, so the x-coordinate r = N / D of φ has nonzero derivative.

theorem TauCeti.Isogeny.wronskian_xRatFunc_sq_mul_Ψ₂Sq {F : Type u_1} [Field F] {W₁ W₂ : WeierstrassCurve.Affine F} [WeierstrassCurve.IsElliptic W₁] [WeierstrassCurve.IsElliptic W₂] {φ : Isogeny W₁ W₂} {c : F} (hc : φ.pullbackDifferential W₂.invariantDifferential = c • W₁.invariantDifferential) : φ.xRatFunc.denom.wronskian φ.xRatFunc.num ^ 2 * WeierstrassCurve.Ψ₂Sq W₁ = Polynomial.C (c ^ 2) * φ.xRatFunc.denom * (Polynomial.C 4 * φ.xRatFunc.num ^ 3 + Polynomial.C (WeierstrassCurve.b₂ W₂) * φ.xRatFunc.num ^ 2 * φ.xRatFunc.denom + Polynomial.C (2 * WeierstrassCurve.b₄ W₂) * φ.xRatFunc.num * φ.xRatFunc.denom ^ 2 + Polynomial.C (WeierstrassCurve.b₆ W₂) * φ.xRatFunc.denom ^ 3)

The Wronskian identity with y eliminated: if φ^*ω₂ = c ω₁, then

w² Ψ₂Sq₁ = c² D (4N³ + b₂'N²D + 2b₄'ND² + b₆'D³),

an identity of polynomials, for N / D = φ.xRatFunc in lowest terms, w its Wronskian, Ψ₂Sq₁ = 4X³ + b₂X² + 2b₄X + b₆ that of W₁, and b₂', b₄', b₆' the invariants of W₂. The last factor is D³ Ψ₂Sq₂(N / D).

theorem TauCeti.Isogeny.eval_wronskian_xRatFunc_sq_mul_eval_Ψ₂Sq {F : Type u_1} [Field F] {W₁ W₂ : WeierstrassCurve.Affine F} [WeierstrassCurve.IsElliptic W₁] [WeierstrassCurve.IsElliptic W₂] {φ : Isogeny W₁ W₂} {c : F} (hc : φ.pullbackDifferential W₂.invariantDifferential = c • W₁.invariantDifferential) {x₀ : F} (hx₀ : Polynomial.eval x₀ φ.xRatFunc.denom ≠ 0) : Polynomial.eval x₀ (φ.xRatFunc.denom.wronskian φ.xRatFunc.num) ^ 2 * Polynomial.eval x₀ (WeierstrassCurve.Ψ₂Sq W₁) = c ^ 2 * Polynomial.eval x₀ φ.xRatFunc.denom ^ 4 * Polynomial.eval (Polynomial.eval x₀ φ.xRatFunc.num / Polynomial.eval x₀ φ.xRatFunc.denom) (WeierstrassCurve.Ψ₂Sq W₂)

Ψ₂Sq₂(r(x₀)) = s(x₀)² Ψ₂Sq₁(x₀), cleared of denominators: if φ^*ω₂ = c ω₁, then at every x₀ where the denominator D of r = φ.xRatFunc does not vanish,

w(x₀)² Ψ₂Sq₁(x₀) = c² D(x₀)⁴ Ψ₂Sq₂(N(x₀) / D(x₀)).

When c ≠ 0 (φ separable), with s = w / (c D²) this is Ψ₂Sq₂(r(x₀)) = s(x₀)² Ψ₂Sq₁(x₀).