Points with singular reduction: the index three case IV*

Let v be a valuation on a field F whose valuation ring O has maximal ideal 𝔪 = (ϖ), 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 c_v is 3 or 1 according to whether the residue quadratic T² + (a₃ / ϖ²) T − a₆ / ϖ⁴ has a root in the residue field.

Tate's algorithm stops at IV* on an equation with ϖ ∣ a₁, ϖ² ∣ a₂, a₃ = ϖ² A₃, ϖ³ ∣ a₄, a₆ = ϖ⁴ A₆ and ϖ⁵ ∤ b₆ = ϖ⁴ (A₃² + 4 A₆), so that the residue quadratic T² + A₃ T − A₆ has distinct roots. A point (x, y) with singular reduction has x, y ∈ 𝔪; comparing powers of ϖ in the Weierstrass equation gives x = ϖ² X and y = ϖ² Y, where the residue of Y is a root of the residue quadratic. The two roots sort the points with singular reduction into two classes, exchanged by negation, since -(x, y) = (x, -y - a₁ x - a₃) and the two roots add up to -A₃.

• If P and Q have singular reduction in different classes, then P + Q has nonsingular reduction. Indeed, if P, Q and P + Q all have singular reduction, then the slope ℓ of the chord through P and Q lies in 𝔪 because ℓ² + a₁ ℓ − a₂ = x₁ + x₂ + x₃, so y₁ − y₂ = ℓ (x₁ − x₂) is divisible by ϖ³, and P and Q are in the same class.
• If P has singular reduction, so does 2 P: the tangent at P has slope (3 x² + 2 a₂ x + a₄ − a₁ y) / (2 y + a₁ x + a₃), whose numerator is divisible by ϖ³ and whose denominator is ϖ² times a unit, as the derivative 2 Y + A₃ of the residue quadratic at the simple root Y is a unit.

So if P has singular reduction, every point lies in E₀(F), in P + E₀(F) or in -P + E₀(F), and these cosets are distinct since 2 P ∉ E₀(F): the index is 3. Otherwise the index is 1. Over a Henselian valuation ring, a root of the residue quadratic lifts to a root Y of T² + A₃ T − A₆, and (0, ϖ² Y) is a point with singular reduction.

In particular [E(F) : E₀(F)] divides 3 with no further hypothesis. No completeness, discreteness, minimality or hypothesis on the residue field is needed except where stated.

Main results

• WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IVStar_of_notMem: at the symbol IV*, if some point has singular reduction, then [E(F) : E₀(F)] = 3.
• WeierstrassCurve.Affine.index_nonsingularReduction_dvd_three_of_isTateTerminal_IVStar: at the symbol IV*, [E(F) : E₀(F)] divides 3.
• WeierstrassCurve.Affine.exists_notMem_nonsingularReduction_of_isTateTerminal_IVStar and WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IVStar: at the symbol IV* over a Henselian valuation ring, if the residue quadratic has a root, some point has singular reduction, and [E(F) : E₀(F)] = 3.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.6.
• 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.

The index at IV*

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IVStar_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.IVStar) {P : W.Point} (hP : P ∉ nonsingularReduction v W) : (nonsingularReduction v W).index = 3

At the reduction symbol IV*, if some point has singular reduction, then [E(F) : E₀(F)] = 3. Here ϖ generates the maximal ideal of the valuation ring, and Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol IV*: ϖ ∣ a₁, ϖ² ∣ a₂, ϖ² ∣ a₃, ϖ³ ∣ a₄, ϖ⁴ ∣ a₆ and ϖ⁵ ∤ b₆. If P has singular reduction, then E₀(F), P + E₀(F) and -P + E₀(F) are the three cosets.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_dvd_three_of_isTateTerminal_IVStar {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.IVStar) : (nonsingularReduction v W).index ∣ 3

At the reduction symbol IV*, [E(F) : E₀(F)] divides 3. It is 3 when some point has singular reduction, and 1 otherwise. Here ϖ generates the maximal ideal of the valuation ring, and Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol IV*.

theorem WeierstrassCurve.Affine.exists_notMem_nonsingularReduction_of_isTateTerminal_IVStar {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.IVStar) {A₃ A₆ : ↥v.valuationSubring} (h₃ : (integralModel (↥v.valuationSubring) W).a₃ = ϖ ^ 2 * A₃) (h₆ : (integralModel (↥v.valuationSubring) W).a₆ = ϖ ^ 4 * A₆) (hroot : ∃ (t : IsLocalRing.ResidueField ↥v.valuationSubring), t ^ 2 + (IsLocalRing.residue ↥v.valuationSubring) A₃ * t - (IsLocalRing.residue ↥v.valuationSubring) A₆ = 0) : ∃ (P : W.Point), P ∉ nonsingularReduction v W

At the reduction symbol IV*, some point has singular reduction when the valuation ring is Henselian, for instance complete, and the residue quadratic T² + A₃ T − A₆ has a root in the residue field, where a₃ = ϖ² A₃ and a₆ = ϖ⁴ A₆. Here ϖ generates the maximal ideal of the valuation ring, and Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol IV*. The root is simple, so it lifts to a root Y of T² + A₃ T − A₆, and the point is (0, ϖ² Y).

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_three_of_isTateTerminal_IVStar {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.IVStar) {A₃ A₆ : ↥v.valuationSubring} (h₃ : (integralModel (↥v.valuationSubring) W).a₃ = ϖ ^ 2 * A₃) (h₆ : (integralModel (↥v.valuationSubring) W).a₆ = ϖ ^ 4 * A₆) (hroot : ∃ (t : IsLocalRing.ResidueField ↥v.valuationSubring), t ^ 2 + (IsLocalRing.residue ↥v.valuationSubring) A₃ * t - (IsLocalRing.residue ↥v.valuationSubring) A₆ = 0) : (nonsingularReduction v W).index = 3

At the reduction symbol IV*, [E(F) : E₀(F)] = 3 when the valuation ring is Henselian, for instance complete, and the residue quadratic T² + A₃ T − A₆ has a root in the residue field, where a₃ = ϖ² A₃ and a₆ = ϖ⁴ A₆. This is the local index c_v = 3 that Tate's algorithm reports at IV* in that case. Here ϖ generates the maximal ideal of the valuation ring, and Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol IV*.