The point-group index reported by Tate's algorithm

Let v be a valuation on a field F, let O be its valuation ring with maximal ideal generated by ϖ, and let W be an integral Weierstrass equation over F. The points with nonsingular reduction form the subgroup E₀(F) of W(F). When Tate's algorithm stops on the integral model, its local index is supposed to equal [W(F) : E₀(F)].

This file proves that equality for the seven additive symbols outside the family Iₙ* with n ≥ 1. At II and II*, every point has nonsingular reduction. At III and III*, the index is 2. At IV and IV*, both indices are 3 precisely when the relevant residue quadratic has a root and are 1 otherwise. The comparison in these two cases uses WeierstrassCurve.hasTateYRoot_iff_exists_residue_root to identify the root test in the definition of WeierstrassCurve.tateLocalIndex with the one used in the point-index theorems. At I₀*, both indices are one more than the number of roots of the residue cubic, and WeierstrassCurve.natCard_tateCubicRoots_eq identifies the roots in O ⧸ (ϖ) counted by the local index with the roots in the residue field counted by the point-index theorem.

It also proves the equality at the symbols Iₙ* with n ≥ 1, where both indices are 4 when the quadratic at which Step 7 of the algorithm stops has a root in the residue field and 2 otherwise; the root tests are compared through WeierstrassCurve.hasTateYRoot_iff_exists_residue_root and WeierstrassCurve.hasTateXRoot_iff_exists_residue_root.

At the multiplicative symbols Iₙ, n ≥ 1, with split tangent quadratic T² + a₁ T − a₂, the index is n, the local index the algorithm reports there. The stopping equation only has its node at the origin modulo ϖ; a translation by elements of the maximal ideal moves the node itself to the origin, so that a₃ = a₄ = 0, and Hensel's lemma lifts the tangent directions. On that centred equation Δ = -a₆ (b₂³ + 432 a₆) with the second factor a unit, so v(a₆) = v(Δ) = v(ϖ)ⁿ, and the index is computed by WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_tangentQuadratic_root. The translation does not change the index, because it is defined over the valuation ring (WeierstrassCurve.Affine.index_nonsingularReduction_baseChange_smul).

For a Henselian discrete valuation ring with perfect residue field this gives the order of E(F) / E₀(F) at split multiplicative reduction intrinsically: on a minimal equation it is the valuation v(Δ_min) of the minimal discriminant (Silverman, The Arithmetic of Elliptic Curves, VII.6.1), since Tate's algorithm stops at I_{v(Δ_min)} after a change of variables over the valuation ring.

At the multiplicative symbols Iₙ with nonsplit tangent quadratic, the local index is gcd(2, n). When n is odd every point has nonsingular reduction: a point reducing to the node would make Δ a unit times a square, while ϖ divides Δ exactly to the odd power n. When n is even and the residue field is finite, the index is 2, by Affine.index_nonsingularReduction_eq_two_of_isTateTerminal_I_of_tangentQuadratic_no_roots. At I₀ the discriminant is a unit and every point has nonsingular reduction. Together with the cases above this proves the equality at every symbol when the valuation ring is Henselian with finite residue field, for instance for the ring of integers of a nonarchimedean local field. The index [W(F) : E₀(F)] is preserved by changes of variables over the valuation ring, so the local index does not depend on the equation on which the algorithm stops.

Main results

• index_nonsingularReduction_eq_tateLocalIndex_of_ne_I_of_ne_IStar_add_one, in the namespace WeierstrassCurve.Affine: at every additive symbol outside Iₙ* with n ≥ 1, the point-group index is the local index reported by Tate's algorithm.
• index_nonsingularReduction_eq_tateLocalIndex_of_isTateTerminal_IStar_add_one, in the namespace WeierstrassCurve.Affine: at Iₙ* with n ≥ 1, the point-group index is the local index reported by Tate's algorithm.
• WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_isTateTerminal_I_of_hasTateTangentRoot: at Iₙ with n ≥ 1 and split tangent quadratic, the point-group index is n.
• WeierstrassCurve.Affine.index_nonsingularReduction_eq_localMinimalDiscriminantValuation: at split multiplicative reduction, the index of E₀ for a minimal equation is the valuation of the minimal discriminant.
• nonsingularReduction_eq_top_of_isTateTerminal_I_of_not_hasTateTangentRoot, in the namespace WeierstrassCurve.Affine: at Iₙ with n odd and nonsplit tangent quadratic, every point has nonsingular reduction.
• WeierstrassCurve.Affine.index_nonsingularReduction_eq_tateLocalIndex: over a Henselian valuation ring with finite residue field, at every symbol, the point-group index is the local index reported by Tate's algorithm.
• WeierstrassCurve.IsTateTerminal.tateLocalIndex_smul: in the same situation, the local index does not depend on the equation on which the algorithm stops.

References

• J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, 2nd ed., VII.6.1.
• 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.Affine.index_nonsingularReduction_eq_tateLocalIndex_of_ne_I_of_ne_IStar_add_one {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] [PerfectField (IsLocalRing.ResidueField ↥v.valuationSubring)] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) {s : TauCeti.ReductionSymbol} (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ s) (hsI : ∀ (n : ℕ), s ≠ TauCeti.ReductionSymbol.I n) (hsIStar : ∀ (n : ℕ), s ≠ TauCeti.ReductionSymbol.IStar (n + 1)) : (nonsingularReduction v W).index = (integralModel (↥v.valuationSubring) W).tateLocalIndex ϖ s

At every additive symbol outside Iₙ* with n ≥ 1, Tate's local index is the point-group index. Suppose that ϖ generates the maximal ideal of the Henselian valuation ring, its residue field is perfect, and Tate's algorithm stops on the integral model at a symbol other than Iₙ or Iₙ* with n ≥ 1. Then the reported local index equals [W(F) : E₀(F)].

The perfectness assumption is used at the symbol III, to lift the double tangent direction; the other six symbols are already covered without it by their individual point-index theorems.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_tateLocalIndex_of_isTateTerminal_IStar_add_one {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}) {n : ℕ} (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ (TauCeti.ReductionSymbol.IStar (n + 1))) : (nonsingularReduction v W).index = (integralModel (↥v.valuationSubring) W).tateLocalIndex ϖ (TauCeti.ReductionSymbol.IStar (n + 1))

At the reduction symbol Iₙ* with n ≥ 1, Tate's local index is the point-group index. Suppose that ϖ generates the maximal ideal of the Henselian valuation ring, and that Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol Iₙ* for some n ≥ 1. Then [W(F) : E₀(F)] is the local index that the algorithm reports: 4 if the quadratic at which the subprocedure of Step 7 stops has a root in the residue field, and 2 otherwise.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_of_isTateTerminal_I_of_hasTateTangentRoot {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}) {n : ℕ} (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ (TauCeti.ReductionSymbol.I (n + 1))) (hroot : (integralModel (↥v.valuationSubring) W).HasTateTangentRoot ϖ) : (nonsingularReduction v W).index = n + 1

At split multiplicative reduction Iₙ, the point-group index is n. Suppose that ϖ generates the maximal ideal of the Henselian valuation ring, that Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol Iₙ for some n ≥ 1, and that the tangent quadratic T² + a₁ T − a₂ has a root modulo ϖ. Then [W(F) : E₀(F)] = n, which is the local index the algorithm reports there (WeierstrassCurve.tateLocalIndex_I_add_one_of_hasTateTangentRoot).

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_localMinimalDiscriminantValuation {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) [DecidableEq F] [IsDiscreteValuationRing ↥v.valuationSubring] [HenselianLocalRing ↥v.valuationSubring] [PerfectField (IsLocalRing.ResidueField ↥v.valuationSubring)] (E : WeierstrassCurve F) [E.IsElliptic] [HasSplitMultiplicativeReduction (↥v.valuationSubring) E] : (nonsingularReduction v E.toAffine).index = localMinimalDiscriminantValuation (↥v.valuationSubring) E

At split multiplicative reduction, [E(F) : E₀(F)] = v(Δ_min) (Silverman, The Arithmetic of Elliptic Curves, VII.6.1). Let the valuation ring of v be a Henselian discrete valuation ring with perfect residue field, and let E be an elliptic curve over F given by a minimal equation with split multiplicative reduction. Then the subgroup E₀(F) of points with nonsingular reduction has index the valuation of the minimal discriminant. Perfectness of the residue field is used only to run Tate's algorithm (WeierstrassCurve.exists_isTateTerminal) on the minimal equation.

theorem WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_isTateTerminal_I_of_not_hasTateTangentRoot {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}) {m : ℕ} (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ (TauCeti.ReductionSymbol.I (2 * m + 1))) (hroot : ¬(integralModel (↥v.valuationSubring) W).HasTateTangentRoot ϖ) : nonsingularReduction v W = ⊤

At nonsplit multiplicative reduction of odd type I₂ₘ₊₁, every point has nonsingular reduction: E₀(F) = W(F). Here ϖ generates the maximal ideal of the valuation ring, Tate's algorithm, run on the integral model with uniformiser ϖ, stops with the symbol I₂ₘ₊₁, and the tangent quadratic T² + a₁ T − a₂ has no root modulo ϖ. This is the nonsplit-node criterion WeierstrassCurve.Affine.nonsingularReduction_eq_top_of_not_splits: Δ is not a unit times a square, since ϖ^(2m+1) divides Δ exactly.

theorem WeierstrassCurve.Affine.index_nonsingularReduction_eq_tateLocalIndex {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] [Finite (IsLocalRing.ResidueField ↥v.valuationSubring)] {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) {s : TauCeti.ReductionSymbol} (h : (integralModel (↥v.valuationSubring) W).IsTateTerminal ϖ s) : (nonsingularReduction v W).index = (integralModel (↥v.valuationSubring) W).tateLocalIndex ϖ s

Tate's local index is the point-group index (Silverman, Advanced Topics, IV.9, Table 4.1). Suppose that the valuation ring of v is Henselian with finite residue field, for instance the ring of integers of a nonarchimedean local field, and that Tate's algorithm, run on the integral model of W with a generator ϖ of the maximal ideal, stops with the symbol s. Then the local index the algorithm reports is [W(F) : E₀(F)], the index of the subgroup of points with nonsingular reduction.

Finiteness of the residue field is used only at nonsplit multiplicative reduction of even type I₂ₘ, to find a point with singular reduction; over a general perfect residue field the index can be 1 there.

theorem WeierstrassCurve.IsTateTerminal.tateLocalIndex_smul {F : Type u_1} {Γ₀ : Type u_2} [Field F] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation F Γ₀) [HenselianLocalRing ↥v.valuationSubring] [Finite (IsLocalRing.ResidueField ↥v.valuationSubring)] {W : WeierstrassCurve ↥v.valuationSubring} {ϖ : ↥v.valuationSubring} (hϖ : IsLocalRing.maximalIdeal ↥v.valuationSubring = Ideal.span {ϖ}) {C : VariableChange ↥v.valuationSubring} {s s' : TauCeti.ReductionSymbol} (h : W.IsTateTerminal ϖ s) (h' : (C • W).IsTateTerminal ϖ s') : (C • W).tateLocalIndex ϖ s' = W.tateLocalIndex ϖ s

Tate's local index does not depend on the equation on which the algorithm stops. Let the valuation ring O of v be Henselian with finite residue field, with maximal ideal generated by ϖ. If Tate's algorithm stops on an equation W over O with the symbol s, and on C • W, for a change of variables C over O, with the symbol s', then the two reported local indices agree: both are the index of the points with nonsingular reduction over the fraction field, which a change of variables over O preserves.