Good reduction from an unramified Tate module

Let K be a nonarchimedean local field with residue field k, let W be an elliptic curve over K given by a minimal equation, and let ℓ be a prime different from the characteristic of k. This file proves the criterion of Néron, Ogg and Shafarevich (Silverman AEC VII.7.1): W has good reduction exactly when the inertia subgroup I_K of Gal(K^{alg}/K) acts trivially on the ℓ-adic Tate module T_ℓ E. Good reduction gives an unramified Tate module by WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_of_mem_inertiaSubgroup; this file proves the converse and assembles the equivalence.

The converse is a counting argument over unramified extensions. Suppose that the reduction is bad.

• Tate's algorithm stops on an integral model of W at a reduction symbol s ≠ I₀.
• Over a finite unramified extension L/K a uniformiser ϖ of K stays a uniformiser, so the algorithm stops at the same symbol s over L (WeierstrassCurve.isTateTerminal_map_iff), and the index [E(L) : E₀(L)] of the points with nonsingular reduction is at most the number of components of s (WeierstrassCurve.Affine.index_nonsingularReduction_le_componentCount), a bound independent of L.
• If E(L) has all n² points of order dividing n, for n prime to the residue characteristic, then n divides that index, because at bad reduction the points of E₀(L)[n] reduce injectively into the n-torsion of the nonsingular points of a singular cubic, of order dividing n (WeierstrassCurve.Affine.Point.dvd_index_nonsingularReduction). This is WeierstrassCurve.le_componentCount_of_natCard_torsionBy_eq.
• If I_K acts trivially on T_ℓ E, every division field K(E[ℓ ^ k]) is unramified over K (WeierstrassCurve.isUnramified_divisionField_pow_of_tateModuleGaloisRepresentation_eq_one), and E has all its ℓ ^ (2k) points of order dividing ℓ ^ k over it. Taking ℓ ^ k larger than the number of components of s contradicts the previous step.

Main results

• WeierstrassCurve.le_componentCount_of_natCard_torsionBy_eq: over a finite unramified extension L/K, if Tate's algorithm stops on the integral model of W at a symbol s ≠ I₀ and E(L) has n² points of order dividing n, with n prime to the residue characteristic, then n is at most the number of components of s.
• WeierstrassCurve.hasGoodReduction_of_tateModuleGaloisRepresentation_eq_one: a minimal equation on whose ℓ-adic Tate module inertia acts trivially has good reduction.
• WeierstrassCurve.hasGoodReduction_iff_tateModuleGaloisRepresentation_eq_one: the criterion of Néron, Ogg and Shafarevich.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.6.1 and VII.7.1.
• J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517.

theorem WeierstrassCurve.le_componentCount_of_natCard_torsionBy_eq {K : Type u_1} {L : Type u_2} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] [Field L] [ValuativeRel L] [TopologicalSpace L] [IsNonarchimedeanLocalField L] [Algebra K L] [ValuativeExtension K L] [TauCeti.IsUnramified K L] [DecidableEq L] {W : WeierstrassCurve K} [IsIntegral (↥(ValuativeRel.valuation K).integer) W] {ϖ : ↥(ValuativeRel.valuation K).integer} (hϖ : Irreducible ϖ) {s : TauCeti.ReductionSymbol} (hs : (integralModel (↥(ValuativeRel.valuation K).integer) W).IsTateTerminal ϖ s) (hs₀ : s ≠ TauCeti.ReductionSymbol.I 0) {n : ℕ} (hn : ↑n ≠ 0) (h : Nat.card ↥(AddSubgroup.torsionBy (W.baseChange L).toAffine.Point ↑n) = n ^ 2) : n ≤ s.componentCount

Full torsion over an unramified extension is bounded at bad reduction (Silverman AEC VII.7.1). Let L/K be an unramified extension of nonarchimedean local fields, and suppose that Tate's algorithm, run with a uniformiser ϖ of K, stops on the integral model of W at a reduction symbol s ≠ I₀, so that W has bad reduction. If E(L) has n² points of order dividing n, for an integer n that is nonzero in the residue field of K, then n is at most the number of components of s.

theorem WeierstrassCurve.hasGoodReduction_of_tateModuleGaloisRepresentation_eq_one {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (W : WeierstrassCurve K) [W.IsElliptic] [IsMinimal (↥(ValuativeRel.valuation K).integer) W] [DecidableEq (AlgebraicClosure K)] {ℓ : ℕ} [Fact (Nat.Prime ℓ)] (hℓ : ↑ℓ ≠ 0) (h : ∀ σ ∈ TauCeti.inertiaSubgroup K, (W.tateModuleGaloisRepresentation ℓ) σ = 1) : HasGoodReduction (↥(ValuativeRel.valuation K).integer) W

An unramified Tate module forces good reduction (Néron, Ogg, Shafarevich; Silverman AEC VII.7.1). Let W be an elliptic curve over a nonarchimedean local field K, given by a minimal equation, and let ℓ be a prime different from the residue characteristic of K. If the inertia subgroup of Gal(K^{alg}/K) acts trivially on the ℓ-adic Tate module of W, then W has good reduction.

theorem WeierstrassCurve.hasGoodReduction_iff_tateModuleGaloisRepresentation_eq_one {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (W : WeierstrassCurve K) [W.IsElliptic] [IsMinimal (↥(ValuativeRel.valuation K).integer) W] [DecidableEq (AlgebraicClosure K)] {ℓ : ℕ} [Fact (Nat.Prime ℓ)] (hℓ : ↑ℓ ≠ 0) : HasGoodReduction (↥(ValuativeRel.valuation K).integer) W ↔ ∀ σ ∈ TauCeti.inertiaSubgroup K, (W.tateModuleGaloisRepresentation ℓ) σ = 1

The criterion of Néron, Ogg and Shafarevich (Silverman AEC VII.7.1). Let W be an elliptic curve over a nonarchimedean local field K, given by a minimal equation, and let ℓ be a prime different from the residue characteristic of K. Then W has good reduction exactly when the inertia subgroup of Gal(K^{alg}/K) acts trivially on the ℓ-adic Tate module of W.