Inertia acts trivially on division points prime to the residue characteristic at good reduction

Let W be a Weierstrass curve over a field K, let L be an extension of K with a valuation v, and suppose that the equation W, read over L, has coefficients in the valuation ring of v. An automorphism σ of L / K in the inertia group of v preserves the valuation ring of v and acts trivially on its residue field, while it fixes the coefficients of W. It therefore does not change the reduction of any point of W(L): the reductions of σ P and P agree (WeierstrassCurve.Affine.Point.reduction_map_of_mem_inertiaSubgroup). At good reduction this says that σ P - P lies in the kernel of the reduction homomorphism.

Now let L be the fraction field of a Dedekind domain A and v the valuation at a height-one prime u of A, and assume good reduction: the integral model of W over the valuation ring has unit discriminant. Let n ∉ u and let P ∈ W(L) be a point such that σ fixes n • P. Then the point σ P - P is killed by n and reduces to the point at infinity, so it is zero, because reduction is injective on torsion of order prime to u (WeierstrassCurve.Affine.Point.injOn_reductionHom_torsionBy). This applies to the n-torsion, where n • P = 0: the inertia group of u acts trivially on W(L)[n], so the n-torsion is unramified at u (Silverman AEC VII.4.1). This is the implication from good reduction to unramified torsion in the criterion of Néron, Ogg and Shafarevich (AEC VII.7.1). It applies equally to the n-division points of the K-rational points, where n • P ∈ W(K): if L is generated over K by the coordinates of such points, as K([n]⁻¹ W(K)) is, then the inertia group of u in the automorphism group of L / K is trivial. When L / K is Galois, as K([n]⁻¹ W(K)) is, this is the group-theoretic form of the unramifiedness of L / K at u (AEC VIII.1.5(b)). This is the input to the unramifiedness of Kummer classes away from the primes of bad reduction and the primes dividing n, which bounds the n-Selmer group.

Main results

• WeierstrassCurve.Affine.Point.reduction_map_of_mem_inertiaSubgroup: an element of the inertia group of v does not change the reduction of a point.
• WeierstrassCurve.Affine.Point.reductionHom_map_of_mem_inertiaSubgroup: the same statement for the reduction homomorphism at good reduction.
• WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup: at good reduction, an element σ of the inertia group of u fixes every point P of W(L) for which it fixes n • P, for an integer n ∉ u.
• WeierstrassCurve.torsionGaloisAction_eq_one_of_mem_inertiaSubgroup: the inertia group of u acts trivially on W(L)[N] for N ∉ u.
• WeierstrassCurve.inertiaSubgroup_eq_bot_of_adjoin_coords_eq_top: a field generated by points Q such that inertia fixes n • Q has trivial inertia group at u in its automorphism group over K.
• WeierstrassCurve.inertiaSubgroup_eq_bot_of_adjoin_coords_eq_top_of_nsmul_eq_baseChange: in particular, so does a field generated by the n-division points of K-rational points.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.4.1, VII.7.1 and VIII.1.5.

theorem WeierstrassCurve.Affine.Point.reduction_map_of_mem_inertiaSubgroup {K : Type u_1} {L : Type u_2} {Γ₀ : Type u_3} [Field K] [Field L] [Algebra K L] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation L Γ₀) {W : WeierstrassCurve K} [IsIntegral (↥v.valuationSubring) (Affine.baseChange W L)] [DecidableEq L] {σ : ↥(ValuationSubring.decompositionSubgroup K v.valuationSubring)} (hσ : σ ∈ ValuationSubring.inertiaSubgroup K v.valuationSubring) (P : (toAffine (Affine.baseChange W L)).Point) : reduction v ((map ↑↑σ) P) = reduction v P

Inertia does not change reductions. An automorphism of L / K in the inertia group of the valuation v sends each point of W(L) to a point with the same reduction modulo v.

theorem WeierstrassCurve.Affine.Point.reductionHom_map_of_mem_inertiaSubgroup {K : Type u_1} {L : Type u_2} {Γ₀ : Type u_3} [Field K] [Field L] [Algebra K L] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation L Γ₀) {W : WeierstrassCurve K} [IsIntegral (↥v.valuationSubring) (Affine.baseChange W L)] [DecidableEq L] [(integralModel (↥v.valuationSubring) (Affine.baseChange W L)).IsElliptic] [DecidableEq (IsLocalRing.ResidueField ↥v.valuationSubring)] {σ : ↥(ValuationSubring.decompositionSubgroup K v.valuationSubring)} (hσ : σ ∈ ValuationSubring.inertiaSubgroup K v.valuationSubring) (P : (toAffine (Affine.baseChange W L)).Point) : (reductionHom v) ((map ↑↑σ) P) = (reductionHom v) P

Inertia does not change reductions, at good reduction. An automorphism of L / K in the inertia group of v commutes with the reduction homomorphism W(L) →+ W_k(k), in the sense that it does not change the image of any point.

theorem WeierstrassCurve.Affine.Point.map_eq_self_of_mem_inertiaSubgroup {A : Type u_1} [CommRing A] [IsDedekindDomain A] {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [Algebra A L] [IsFractionRing A L] (u : IsDedekindDomain.HeightOneSpectrum A) {W : WeierstrassCurve K} [IsIntegral (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (Affine.baseChange W L)] [(integralModel (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (Affine.baseChange W L)).IsElliptic] [DecidableEq L] {n : ℕ} (hn : ↑n ∉ u.asIdeal) {σ : ↥(ValuationSubring.decompositionSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring)} (hσ : σ ∈ ValuationSubring.inertiaSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) {P : (toAffine (Affine.baseChange W L)).Point} (hP : (map ↑↑σ) (n • P) = n • P) : (map ↑↑σ) P = P

Division points are unramified at good reduction (Silverman AEC VII.4.1 and VIII.1.5(b)): if W has good reduction at u, an automorphism σ of L / K in the inertia group of u fixes every point P of W(L) such that σ fixes n • P, for an integer n ∉ u. Taking n • P = 0, it fixes the n-torsion of W(L); taking n • P in W(K), it fixes the n-division points of the K-rational points.

theorem WeierstrassCurve.torsionGaloisAction_eq_one_of_mem_inertiaSubgroup {A : Type u_1} [CommRing A] [IsDedekindDomain A] {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [Algebra A L] [IsFractionRing A L] (u : IsDedekindDomain.HeightOneSpectrum A) (W : WeierstrassCurve K) [IsIntegral (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)] [(integralModel (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)).IsElliptic] [DecidableEq L] {N : ℤ} (hN : ↑N ∉ u.asIdeal) {σ : ↥(ValuationSubring.decompositionSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring)} (hσ : σ ∈ ValuationSubring.inertiaSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) : (W.torsionGaloisAction N) ↑σ = 1

The N-torsion is unramified at good reduction (Silverman AEC VII.4.1): if W has good reduction at u and N ∉ u, every element of the inertia group of u acts trivially on W(L)[N].

theorem WeierstrassCurve.inertiaSubgroup_eq_bot_of_adjoin_coords_eq_top {A : Type u_1} [CommRing A] [IsDedekindDomain A] {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [Algebra A L] [IsFractionRing A L] (u : IsDedekindDomain.HeightOneSpectrum A) {W : WeierstrassCurve K} [IsIntegral (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)] [(integralModel (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)).IsElliptic] [DecidableEq L] {n : ℕ} (hn : ↑n ∉ u.asIdeal) {S : Set (W.baseChange L).toAffine.Point} (hS : ∀ σ ∈ ValuationSubring.inertiaSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring, ∀ Q ∈ S, (Affine.Point.map ↑↑σ) (n • Q) = n • Q) (hL : IntermediateField.adjoin K (Affine.Point.xCoord '' S ∪ Affine.Point.yCoord '' S) = ⊤) : ValuationSubring.inertiaSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring = ⊥

A field generated by points with inertia-fixed multiples has trivial inertia at good reduction. Let S be a set of points Q of W(L) such that every element of the inertia group of u fixes n • Q, and suppose that their coordinates generate L over K. If W has good reduction at u and n ∉ u, the inertia group of u in the automorphism group of L / K is trivial. For the n-division points of K-rational points, see WeierstrassCurve.inertiaSubgroup_eq_bot_of_adjoin_coords_eq_top_of_nsmul_eq_baseChange.

theorem WeierstrassCurve.inertiaSubgroup_eq_bot_of_adjoin_coords_eq_top_of_nsmul_eq_baseChange {A : Type u_1} [CommRing A] [IsDedekindDomain A] {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [Algebra A L] [IsFractionRing A L] (u : IsDedekindDomain.HeightOneSpectrum A) {W : WeierstrassCurve K} [IsIntegral (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)] [(integralModel (↥(IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring) (W.baseChange L)).IsElliptic] [DecidableEq L] [DecidableEq K] {n : ℕ} (hn : ↑n ∉ u.asIdeal) {S : Set (W.baseChange L).toAffine.Point} (hS : ∀ Q ∈ S, ∃ (P : W.toAffine.Point), n • Q = (Affine.Point.baseChange K L) P) (hL : IntermediateField.adjoin K (Affine.Point.xCoord '' S ∪ Affine.Point.yCoord '' S) = ⊤) : ValuationSubring.inertiaSubgroup K (IsDedekindDomain.HeightOneSpectrum.valuation L u).valuationSubring = ⊥

The field of division points has trivial inertia at good reduction. Let S be a set of n-division points of K-rational points of W, that is points Q of W(L) with n • Q in the image of W(K), and suppose that their coordinates generate L over K, as they do for L = K([n]⁻¹ W(K)). If W has good reduction at u and n ∉ u, the inertia group of u in the automorphism group of L / K is trivial. This group detects ramification only when L / K is Galois, as it is for L = K([n]⁻¹ W(K)), where this is Silverman AEC VIII.1.5(b).