Points with singular reduction at the symbol IV

Let v be a valuation on a field F, with valuation ring O, maximal ideal 𝔪 = (ϖ) and residue field k, and let W be a Weierstrass curve over F with an integral model W_O over O. The points with nonsingular reduction form the subgroup E₀(F) of W(F) (WeierstrassCurve.Affine.nonsingularReduction). When F is complete for a discrete valuation with perfect residue field and the equation is minimal, the index [E(F) : E₀(F)] is the local index c_v that Tate's algorithm computes (Silverman, Advanced Topics, IV.9, Table 4.1). This file computes the index at the reduction symbol IV, where Tate's algorithm stops when ϖ ∣ b₂, ϖ ∣ a₃, ϖ ∣ a₄, ϖ² ∣ a₆, ϖ³ ∣ b₈ and ϖ³ ∤ b₆. Writing a₃ = ϖ A₃ and a₆ = ϖ² A₆, the algorithm reports c_v = 3 when the residue quadratic T² + A₃ T − A₆ has a root in k, and c_v = 1 otherwise. The last stopping condition says that this quadratic has the unit discriminant A₃² + 4 A₆, so its roots are distinct.

The reduced curve has a cusp at the origin, whose tangent cone is the double line y = s x, for s a lift of the double root of the tangent quadratic T² + a₁ T − a₂. Such a lift exists over any residue field: s = −a₁ / 2 when 2 is a unit, and s = a₄ / a₃ when a₃ / ϖ is a unit, which is the case when 2 ∈ 𝔪 because the discriminant A₃² + 4 A₆ is a unit. The condition ϖ³ ∣ b₈ then says that a₄ / ϖ ≡ s a₃ / ϖ modulo 𝔪.

A point with singular reduction is (ϖ X, ϖ Y) with X, Y ∈ O, and the Weierstrass equation divided by ϖ² says that Y − s X reduces to a root of the residue quadratic. Negation exchanges the two roots. The slope ℓ of the chord through two points P and Q with singular reduction, when P + Q also has singular reduction, is integral with ℓ ≡ s modulo 𝔪, so Y₁ − Y₂ = ℓ (X₁ − X₂) shows that P and Q reduce to the same root. The tangent at a point with singular reduction has slope ≡ s as well, its denominator being ϖ times the unit 2 (Y − s X) + A₃ up to 𝔪, so 2 P again has singular reduction. Hence, if P has singular reduction, the three cosets of E₀(F) are E₀(F), P + E₀(F) and −P + E₀(F), and the index is 3. Over a Henselian valuation ring a root t of the residue quadratic lifts to a root Y of T² + A₃ T − A₆, and (0, ϖ Y) is a point with singular reduction.

No discreteness, completeness, minimality or perfectness is assumed. Only the existence of a point with singular reduction uses the Henselian hypothesis.

Main results

• WeierstrassCurve.dvd_residueQuadratic_of_equation: over a domain, with a₃ = ϖ A₃, a₄ = ϖ A₄, a₆ = ϖ² A₆ and s a lift of the double root of the tangent quadratic with A₄ ≡ s A₃, a solution (ϖ X, ϖ Y) makes Y − s X a root of T² + A₃ T − A₆ modulo ϖ.
• WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_isTateTerminal_IV: at IV, if the residue quadratic has no root in the residue field, every point has nonsingular reduction.
• WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IV_of_notMem: at IV, if some point has singular reduction, then [E(F) : E₀(F)] = 3.
• WeierstrassCurve.Affine.index_nonsingularReduction_dvd_three_of_isTateTerminal_IV: at IV, the index divides 3.
• WeierstrassCurve.Affine.exists_notMem_nonsingularReduction_of_isTateTerminal_IV and WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IV: at IV over a Henselian valuation ring, if the residue quadratic has a root in the residue field, some point has singular reduction, and [E(F) : E₀(F)] = 3.

References

• J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, IV.9.
• 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.dvd_residueQuadratic_of_equation {R : Type u_1} [CommRing R] [IsDomain R] (W : WeierstrassCurve R) {ϖ A₃ A₄ A₆ s : R} (hϖ : ϖ ≠ 0) (h₃ : W.a₃ = ϖ * A₃) (h₄ : W.a₄ = ϖ * A₄) (h₆ : W.a₆ = ϖ ^ 2 * A₆) (hs₁ : ϖ ∣ 2 * s + W.a₁) (hs₂ : ϖ ∣ s ^ 2 + W.a₁ * s - W.a₂) (hs₄ : ϖ ∣ A₄ - s * A₃) {X Y : R} (h : W.toAffine.Equation (ϖ * X) (ϖ * Y)) : ϖ ∣ (Y - s * X) ^ 2 + A₃ * (Y - s * X) - A₆

On an equation with a₃ = ϖ A₃, a₄ = ϖ A₄ and a₆ = ϖ² A₆, let s lift a double root of the tangent quadratic modulo ϖ, with A₄ ≡ s A₃. Then a solution (ϖ X, ϖ Y) makes Y − s X a root of the residue quadratic T² + A₃ T − A₆ modulo ϖ.

Numerals

The data of the stopping test

Points with singular reduction

The index at the symbol IV

theorem WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_isTateTerminal_IV {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) {W : Affine F} [IsIntegral (↥v.valuationSubring) W] [DecidableEq F] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ TauCeti.ReductionSymbol.IV) {A₃ A₆ : ↥v.valuationSubring} (h₃ : (integralModel (↥v.valuationSubring) W).a₃ = ϖ * A₃) (h₆ : (integralModel (↥v.valuationSubring) W).a₆ = ϖ ^ 2 * A₆) (hroot : ∀ (t : IsLocalRing.ResidueField ↥v.valuationSubring), t ^ 2 + (IsLocalRing.residue ↥v.valuationSubring) A₃ * t - (IsLocalRing.residue ↥v.valuationSubring) A₆ ≠ 0) : nonsingularReduction v W = ⊤

At the symbol IV with no rational root of the residue quadratic, every point has nonsingular reduction: E₀(F) = W(F). Here ϖ generates the maximal ideal, Tate's algorithm stops at IV on the integral model, and with a₃ = ϖ A₃ and a₆ = ϖ² A₆ the residue quadratic T² + A₃ T − A₆ has no root in the residue field.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IV_of_notMem {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) {W : Affine F} [IsIntegral (↥v.valuationSubring) W] [DecidableEq F] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ TauCeti.ReductionSymbol.IV) {P : W.Point} (hP : P ∉ nonsingularReduction v W) : (nonsingularReduction v W).index = 3

At the symbol IV, if some point has singular reduction, then [E(F) : E₀(F)] = 3. Here ϖ generates the maximal ideal and Tate's algorithm stops at IV on the integral model. If P has singular reduction, the cosets of E₀(F) are E₀(F), P + E₀(F) and −P + E₀(F).

theorem WeierstrassCurve.Affine.index_nonsingularReduction_dvd_three_of_isTateTerminal_IV {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) {W : Affine F} [IsIntegral (↥v.valuationSubring) W] [DecidableEq F] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ TauCeti.ReductionSymbol.IV) : (nonsingularReduction v W).index ∣ 3

At the symbol IV, [E(F) : E₀(F)] divides 3. Here ϖ generates the maximal ideal and Tate's algorithm stops at IV on the integral model. The index is 3 when some point has singular reduction, and 1 otherwise.

theorem WeierstrassCurve.Affine.exists_notMem_nonsingularReduction_of_isTateTerminal_IV {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) {W : Affine F} [IsIntegral (↥v.valuationSubring) W] [DecidableEq F] [HenselianLocalRing ↥v.valuationSubring] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ TauCeti.ReductionSymbol.IV) {A₃ A₆ : ↥v.valuationSubring} (h₃ : (integralModel (↥v.valuationSubring) W).a₃ = ϖ * A₃) (h₆ : (integralModel (↥v.valuationSubring) W).a₆ = ϖ ^ 2 * A₆) {t : IsLocalRing.ResidueField ↥v.valuationSubring} (ht : t ^ 2 + (IsLocalRing.residue ↥v.valuationSubring) A₃ * t - (IsLocalRing.residue ↥v.valuationSubring) A₆ = 0) : ∃ (P : W.Point), P ∉ nonsingularReduction v W

At the symbol IV, some point has singular reduction, when the valuation ring is Henselian, for instance complete, and the residue quadratic has a root. Here ϖ generates the maximal ideal, Tate's algorithm stops at IV on the integral model, and with a₃ = ϖ A₃ and a₆ = ϖ² A₆ the residue quadratic T² + A₃ T − A₆ has a root t in the residue field. The root is simple, so it lifts to a root Y of T² + A₃ T − A₆, and the point is (0, ϖ Y).

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IV {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) {W : Affine F} [IsIntegral (↥v.valuationSubring) W] [DecidableEq F] [HenselianLocalRing ↥v.valuationSubring] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ TauCeti.ReductionSymbol.IV) {A₃ A₆ : ↥v.valuationSubring} (h₃ : (integralModel (↥v.valuationSubring) W).a₃ = ϖ * A₃) (h₆ : (integralModel (↥v.valuationSubring) W).a₆ = ϖ ^ 2 * A₆) {t : IsLocalRing.ResidueField ↥v.valuationSubring} (ht : t ^ 2 + (IsLocalRing.residue ↥v.valuationSubring) A₃ * t - (IsLocalRing.residue ↥v.valuationSubring) A₆ = 0) : (nonsingularReduction v W).index = 3

At the symbol IV, [E(F) : E₀(F)] = 3, when the valuation ring is Henselian, for instance complete, and the residue quadratic has a root. This is the local index c_v = 3 that Tate's algorithm reports at IV in that case. Here ϖ generates the maximal ideal, Tate's algorithm stops at IV on the integral model, and with a₃ = ϖ A₃ and a₆ = ϖ² A₆ the residue quadratic T² + A₃ T − A₆ has a root t in the residue field.