Good reduction from unramified torsion at potential good reduction

Let K be a nonarchimedean local field with residue field k, and let W be an elliptic curve over K given by a minimal equation, with potential good reduction. This file proves the finite-level form of the criterion of Néron, Ogg and Shafarevich (Serre–Tate; compare Silverman AEC VII.7.1 and VII.7.3): for an integer N ≥ 3 prime to the characteristic of k, the curve has good reduction exactly when the inertia subgroup I_K of Gal(K^{alg}/K) fixes every point of E(K^{alg})[N], equivalently when the division field K(E[N]) is unramified over K.

Good reduction fixes the division points prime to char k (WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup_algebraicClosure). For the converse, write ℓ ^ k for a divisor of N equal to 4 or to an odd prime, so that ℓ ^ k ≥ 3.

• At potential good reduction, I_K acts on T_ℓ E through a finite quotient (WeierstrassCurve.HasPotentialGoodReduction.finite_image_tateModuleGaloisRepresentation).
• An element of I_K fixes E[ℓ ^ k] ⊆ E[N], so by Serre's lemma it acts trivially on T_ℓ E (WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_of_finite_image).
• An unramified Tate module forces good reduction (WeierstrassCurve.hasGoodReduction_of_tateModuleGaloisRepresentation_eq_one).

The hypothesis of potential good reduction cannot be dropped, since a curve with multiplicative reduction can have unramified N-torsion. For instance the curve 11a1 has split multiplicative reduction at 11 with minimal discriminant of valuation 5, so its Tate parameter is a fifth power up to an unramified unit and its 5-torsion is unramified at 11.

Main results

• WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_of_forall_mem_inertiaSubgroup: at potential good reduction, if I_K fixes E(K^{alg})[N] for some N ≥ 3 prime to char k, then W has good reduction.
• WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_iff_forall_mem_inertiaSubgroup: at potential good reduction, W has good reduction exactly when I_K fixes E(K^{alg})[N].
• WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_iff_isUnramified_divisionField: at potential good reduction, W has good reduction exactly when K(E[N])/K is unramified.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.7.1 and VII.7.3.
• J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492–517.

theorem WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_of_forall_mem_inertiaSubgroup {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] {W : WeierstrassCurve K} [W.IsElliptic] [IsMinimal (↥(ValuativeRel.valuation K).integer) W] (h : HasPotentialGoodReduction (↥(ValuativeRel.valuation K).integer) W) {N : ℕ} (hN : 3 ≤ N) (hNk : ↑N ≠ 0) (hfix : ∀ σ ∈ TauCeti.inertiaSubgroup K, ∀ (P : (W.baseChange (AlgebraicClosure K)).toAffine.Point), N • P = 0 → (Affine.Point.map ↑σ) P = P) : HasGoodReduction (↥(ValuativeRel.valuation K).integer) W

Unramified torsion forces good reduction at potential good reduction (Néron, Ogg, Shafarevich; Serre–Tate). Let W be an elliptic curve over a nonarchimedean local field K, given by a minimal equation, with potential good reduction. If the inertia subgroup of Gal(K^{alg}/K) fixes every point of E(K^{alg})[N], for an integer N ≥ 3 that is nonzero in the residue field of K, then W has good reduction.

theorem WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_iff_forall_mem_inertiaSubgroup {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] {W : WeierstrassCurve K} [W.IsElliptic] [IsMinimal (↥(ValuativeRel.valuation K).integer) W] (h : HasPotentialGoodReduction (↥(ValuativeRel.valuation K).integer) W) {N : ℕ} (hN : 3 ≤ N) (hNk : ↑N ≠ 0) : HasGoodReduction (↥(ValuativeRel.valuation K).integer) W ↔ ∀ σ ∈ TauCeti.inertiaSubgroup K, ∀ (P : (W.baseChange (AlgebraicClosure K)).toAffine.Point), N • P = 0 → (Affine.Point.map ↑σ) P = P

The criterion of Néron, Ogg and Shafarevich with finite coefficients (Serre–Tate). Let W be an elliptic curve over a nonarchimedean local field K, given by a minimal equation, with potential good reduction, and let N ≥ 3 be an integer that is nonzero in the residue field of K. Then W has good reduction exactly when the inertia subgroup of Gal(K^{alg}/K) fixes every point of E(K^{alg})[N].

theorem WeierstrassCurve.HasPotentialGoodReduction.hasGoodReduction_iff_isUnramified_divisionField {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] {W : WeierstrassCurve K} [W.IsElliptic] [IsMinimal (↥(ValuativeRel.valuation K).integer) W] (h : HasPotentialGoodReduction (↥(ValuativeRel.valuation K).integer) W) {N : ℕ} (hN : 3 ≤ N) (hNk : ↑N ≠ 0) [ValuativeRel ↥(W.divisionField N)] [TopologicalSpace ↥(W.divisionField N)] [IsNonarchimedeanLocalField ↥(W.divisionField N)] [ValuativeExtension K ↥(W.divisionField N)] : HasGoodReduction (↥(ValuativeRel.valuation K).integer) W ↔ TauCeti.IsUnramified K ↥(W.divisionField N)

Good reduction is an unramified N-division field at potential good reduction (Serre–Tate). Let W be an elliptic curve over a nonarchimedean local field K, given by a minimal equation, with potential good reduction, and let N ≥ 3 be an integer that is nonzero in the residue field of K. Then W has good reduction exactly when the N-division field K(E[N]) is unramified over K, for every structure of nonarchimedean local field on it compatible with K.