The discriminant in the cubic normal form of Tate's algorithm

In the normal form with ϖ ∣ a₁, a₂, ϖ² ∣ a₃, a₄ and ϖ³ ∣ a₆, write a₂ = ϖ A₂, a₄ = ϖ² A₄ and a₆ = ϖ³ A₆. The discriminant is divisible by ϖ⁶, and its quotient reduces to sixteen times the discriminant of the residue cubic T³ + A₂ T² + A₄ T + A₆. Consequently, its valuation is exactly six if and only if the residue characteristic is not two and that cubic has nonzero discriminant.

This connects the cubic test of Tate's algorithm with the discriminant valuation used in its algorithmic Ogg exponent. In residue characteristic two the quotient always vanishes; a cubic with distinct roots therefore need not give discriminant valuation six. No perfectness, Henselianity or minimality assumption is needed for these discriminant computations.

Main results

• WeierstrassCurve.exists_Δ_eq_pow_six_mul: the factorization and congruence over any commutative ring.
• WeierstrassCurve.residue_Δ_div_pow_six: the residue of Δ / ϖ⁶ over a local ring.
• WeierstrassCurve.six_le_addVal_Δ_of_pow_dvd: the valuation bound in the cubic normal form.
• WeierstrassCurve.addVal_Δ_eq_six_iff: the exact valuation test, in every residue characteristic.

References

• J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, IV.9, Step 6 of Tate's algorithm.
• J. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil, in Modular Functions of One Variable IV, LNM 476 (1975), 33–52.

theorem WeierstrassCurve.exists_Δ_eq_pow_six_mul {R : Type u_1} [CommRing R] (W : WeierstrassCurve R) (ϖ A₂ A₄ A₆ : R) (h₁ : ϖ ∣ W.a₁) (h₂ : W.a₂ = ϖ * A₂) (h₃ : ϖ ^ 2 ∣ W.a₃) (h₄ : W.a₄ = ϖ ^ 2 * A₄) (h₆ : W.a₆ = ϖ ^ 3 * A₆) : ∃ (d : R), W.Δ = ϖ ^ 6 * d ∧ ϖ ∣ d - 16 * { a := 1, b := A₂, c := A₄, d := A₆ }.discr

In the cubic normal form, Δ = ϖ⁶ d where d is congruent modulo ϖ to sixteen times the discriminant of T³ + A₂ T² + A₄ T + A₆. This identity holds over any commutative ring.

theorem WeierstrassCurve.pow_six_dvd_Δ_of_pow_dvd {R : Type u_1} [CommRing R] (W : WeierstrassCurve R) (ϖ : R) (h₁ : ϖ ∣ W.a₁) (h₂ : ϖ ∣ W.a₂) (h₃ : ϖ ^ 2 ∣ W.a₃) (h₄ : ϖ ^ 2 ∣ W.a₄) (h₆ : ϖ ^ 3 ∣ W.a₆) : ϖ ^ 6 ∣ W.Δ

The discriminant in the cubic normal form is divisible by the sixth power of the parameter. The parameter need not be a uniformiser or even a nonzero element.

theorem WeierstrassCurve.residue_Δ_div_pow_six {R : Type u_1} [CommRing R] [IsLocalRing R] {ϖ : R} (hϖ : ϖ ∈ IsLocalRing.maximalIdeal R) (hϖ₆ : ϖ ^ 6 ≠ 0) (W : WeierstrassCurve R) (A₂ A₄ A₆ d : R) (h₁ : ϖ ∣ W.a₁) (h₂ : W.a₂ = ϖ * A₂) (h₃ : ϖ ^ 2 ∣ W.a₃) (h₄ : W.a₄ = ϖ ^ 2 * A₄) (h₆ : W.a₆ = ϖ ^ 3 * A₆) (hd : W.Δ = ϖ ^ 6 * d) : (IsLocalRing.residue R) d = 16 * { a := 1, b := (IsLocalRing.residue R) A₂, c := (IsLocalRing.residue R) A₄, d := (IsLocalRing.residue R) A₆ }.discr

The residue of any quotient Δ / ϖ⁶ is sixteen times the discriminant of the residue cubic. The parameter ϖ only has to lie in the maximal ideal with ϖ⁶ ≠ 0 (for instance a nonzero non-unit of a local domain), and the statement is independent of the choices of divisible-coefficient witnesses.

theorem WeierstrassCurve.six_le_addVal_Δ_of_pow_dvd {R : Type u_1} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] {ϖ : R} (hϖ : ϖ ∈ IsLocalRing.maximalIdeal R) (W : WeierstrassCurve R) (h₁ : ϖ ∣ W.a₁) (h₂ : ϖ ∣ W.a₂) (h₃ : ϖ ^ 2 ∣ W.a₃) (h₄ : ϖ ^ 2 ∣ W.a₄) (h₆ : ϖ ^ 3 ∣ W.a₆) : 6 ≤ (IsDiscreteValuationRing.addVal R) W.Δ

The discriminant valuation in the cubic normal form is at least six. The parameter ϖ may be any non-unit; it need not be a uniformiser.

theorem WeierstrassCurve.addVal_Δ_eq_six_iff {R : Type u_1} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] {ϖ : R} (hϖ : Irreducible ϖ) (W : WeierstrassCurve R) (A₂ A₄ A₆ : R) (h₁ : ϖ ∣ W.a₁) (h₂ : W.a₂ = ϖ * A₂) (h₃ : ϖ ^ 2 ∣ W.a₃) (h₄ : W.a₄ = ϖ ^ 2 * A₄) (h₆ : W.a₆ = ϖ ^ 3 * A₆) : (IsDiscreteValuationRing.addVal R) W.Δ = 6 ↔ 2 ≠ 0 ∧ { a := 1, b := (IsLocalRing.residue R) A₂, c := (IsLocalRing.residue R) A₄, d := (IsLocalRing.residue R) A₆ }.discr ≠ 0

The discriminant valuation in the cubic normal form is exactly six precisely when two is nonzero in the residue field and the residue cubic has nonzero discriminant.