Points with singular reduction at a split node

Let v be a valuation on a field F, with valuation ring O and maximal ideal 𝔪, and let W be an elliptic curve over F with an integral model whose reduction is a split node at the origin. Centred at the node, as a translation over a Henselian O always allows, the model has a₃ = a₄ = 0, a₆ ∈ 𝔪, b₂ a unit, and tangent quadratic T² + a₁ T - a₂ = (T - α) (T - β) with α, β ∈ O and β - α a unit. In the branch coordinates u = y - α x and w = y - β x the equation reads u w = x³ + a₆, and the points with singular reduction are the affine points with u, w ∈ 𝔪. This file computes the index of the subgroup E₀(F) of points with nonsingular reduction (WeierstrassCurve.Affine.nonsingularReduction) when O is Henselian:

[E(F) : E₀(F)] = #{γ ∈ v(F) | v a₆ < γ ≤ 1}.

When 𝔪 = (ϖ) and v a₆ = v ϖ ^ n, which is v Δ = v ϖ ^ n, the index is n: this is the local index c_v = n that Tate's algorithm reports at split multiplicative reduction of type Iₙ (Silverman, Advanced Topics, IV.9, Table 4.1), and the order v Δ of E(K) / E₀(K) in Silverman VII.6.1, which also shows that this group is cyclic; only its order is computed here. Neither completeness nor discreteness of v is needed for the first formula.

The cosets of E₀(F) are read off a single value of v attached to each point P = (x, y) outside E₀(F): v u when v w ≤ v u, and v a₆ / v w when v u < v w. Both lie in (v a₆, 1). Three facts make this value a bijection from the nontrivial cosets onto the values of v in that interval. Each is a valuation estimate along the line y = ℓ x + ν through two points, which meets the curve again at the third point of the chord, with A = ℓ - α and B = ℓ - β: the branch coordinates of the three points are uᵢ = A xᵢ + ν and wᵢ = B xᵢ + ν, and x₁ + x₂ + x₃ = A B, x₁ x₂ + x₁ x₃ + x₂ x₃ = -(A + B) ν.

• The value is constant on cosets: if P has v w₁ ≤ v u₁ and Q has nonsingular reduction, then A and B share a valuation σ ≥ 1 with v u₂ = v w₂ = σ³, and in w₂ w₃ = w₁² - 3 B x₁ w₁ + 3 B² x₁² - B³ u₁ the last term dominates, so v w₃ = v u₁.
• The value separates cosets: if v w₁ ≤ v u₁ = v w₂ and v u₂ ≤ v w₂, then in u₃ = A² B + 3 ν - u₁ - u₂ the first term dominates and u₃ ∉ 𝔪, so P + Q ∈ E₀(F).
• Every value occurs: for v a₆ ≤ (v z)² < 1 the equation u w = x³ + a₆ with u = z and w = z t becomes t = z (1 - t)³ / (β - α)³ + a₆ / z², which Hensel's lemma solves.

Main results

• WeierstrassCurve.Affine.index_nonsingularReduction_eq_natCard_of_tangentQuadratic_root: over a Henselian valuation ring, [E(F) : E₀(F)] is the number of values of v in (v a₆, 1].
• WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_tangentQuadratic_root: if moreover the maximal ideal is generated by ϖ and v a₆ = v ϖ ^ n with n > 0, then [E(F) : E₀(F)] = n.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.2 and 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.

Lines meeting the cubic (y - α x) (y - β x) = x³ + a₆

Valuations along such a line

Points of a curve in split nodal normal form

The component of a point

The index of E₀(F)

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_natCard_of_tangentQuadratic_root {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] [WeierstrassCurve.IsElliptic W] (h₃ : W.a₃ = 0) (h₄ : W.a₄ = 0) (h₆ : v W.a₆ < 1) (hb₂ : v (b₂ W) = 1) {α : F} (hαr : α ^ 2 + W.a₁ * α - W.a₂ = 0) : (nonsingularReduction v W).index = Nat.card ↑(Set.range ⇑v ∩ Set.Ioc (v W.a₆) 1)

The index of E₀(F) at a split node is the number of values of v in (v a₆, 1]. Let v be a valuation on F whose valuation ring is Henselian, and W an elliptic curve over F with an integral model whose reduction is a split node at the origin, centred so that a₃ = a₄ = 0: a₆ lies in the maximal ideal, b₂ is a unit, and the tangent quadratic T² + a₁ T - a₂ has a root α in F (necessarily in the valuation ring). Then the index of the subgroup E₀(F) of points with nonsingular reduction is the number of values γ of v with v a₆ < γ ≤ 1, both sides being 0 when infinite.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_tangentQuadratic_root {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] [WeierstrassCurve.IsElliptic W] (h₃ : W.a₃ = 0) (h₄ : W.a₄ = 0) (hb₂ : v (b₂ W) = 1) {α : F} (hαr : α ^ 2 + W.a₁ * α - W.a₂ = 0) {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) {n : ℕ} (hn : 0 < n) (h₆ : v W.a₆ = v ↑ϖ ^ n) : (nonsingularReduction v W).index = n

The local index at split multiplicative reduction Iₙ is n. Let v be a valuation on F whose valuation ring is Henselian with maximal ideal generated by ϖ, and W an elliptic curve over F with an integral model, centred at the singular point of its reduction so that a₃ = a₄ = 0, with b₂ a unit, v a₆ = v ϖ ^ n for some n > 0, and tangent quadratic T² + a₁ T - a₂ with a root α in F (necessarily in the valuation ring). Then the points with nonsingular reduction have index [E(F) : E₀(F)] = n.