Unramified torsion and unramified division fields

Let W be a Weierstrass curve over a nonarchimedean local field K, with separable closure Kˢ and inertia subgroup I_K. The m-division field K(E[m]) ⊆ Kˢ is generated by the coordinates of the points of E(Kˢ)[m], and the subgroup of Gal(Kˢ/K) fixing it is the kernel of the action on E(Kˢ)[m]. Hence K(E[m])/K is unramified exactly when inertia acts trivially on E(Kˢ)[m]: the torsion module E[m] is unramified exactly when the field it cuts out is (Silverman, VII.4 and VII.7).

If inertia acts trivially on the Tate module T_ℓ E for a prime ℓ invertible in K, it acts trivially on every level E[ℓ ^ k], since every point of E[ℓ ^ k] is a component of a point of T_ℓ E (WeierstrassCurve.tateModuleGaloisRepresentation_eq_one_iff). So each K(E[ℓ ^ k]) is a finite unramified extension of K, over which E has all ℓ ^ (2k) points of order dividing ℓ ^ k (WeierstrassCurve.natCard_torsionBy_of_divisionField_le). This is the step of the criterion of Néron, Ogg and Shafarevich (Silverman, VII.7.1) that turns an unramified Tate module into rational torsion over unramified extensions of K.

Main results

• WeierstrassCurve.isUnramified_divisionField_iff: K(E[m])/K is unramified exactly when inertia fixes every point of E(Kˢ)[m].
• WeierstrassCurve.isUnramified_divisionField_iff_forall_mem_inertiaSubgroup: for W elliptic and m invertible in K, equivalently inertia fixes every point of E(K^{alg})[m].
• WeierstrassCurve.isUnramified_divisionField_pow_of_tateModuleGaloisRepresentation_eq_one: if inertia acts trivially on T_ℓ E, then K(E[ℓ ^ k])/K is unramified for every k.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.4 and VII.7.

theorem WeierstrassCurve.isUnramified_divisionField_iff {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (W : WeierstrassCurve K) (m : ℕ) [ValuativeRel ↥(W.divisionField m)] [TopologicalSpace ↥(W.divisionField m)] [IsNonarchimedeanLocalField ↥(W.divisionField m)] [ValuativeExtension K ↥(W.divisionField m)] : TauCeti.IsUnramified K ↥(W.divisionField m) ↔ ∀ σ ∈ Subgroup.map (TauCeti.absoluteGaloisGroupRestrictEquiv K).toMonoidHom (TauCeti.inertiaSubgroup K), ∀ (P : W.TorsionCoeff m), σ • P = P

The m-division field is unramified exactly when inertia acts trivially on E[m]. For a structure of nonarchimedean local field on K(E[m]) compatible with K, the extension K(E[m])/K is unramified exactly when every element of the inertia subgroup, read in Gal(Kˢ/K), fixes every point of E(Kˢ)[m].

theorem WeierstrassCurve.isUnramified_divisionField_iff_forall_mem_inertiaSubgroup {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (W : WeierstrassCurve K) [W.IsElliptic] {m : ℕ} (hm : ↑m ≠ 0) [ValuativeRel ↥(W.divisionField m)] [TopologicalSpace ↥(W.divisionField m)] [IsNonarchimedeanLocalField ↥(W.divisionField m)] [ValuativeExtension K ↥(W.divisionField m)] : TauCeti.IsUnramified K ↥(W.divisionField m) ↔ ∀ σ ∈ TauCeti.inertiaSubgroup K, ∀ (P : (W.baseChange (AlgebraicClosure K)).toAffine.Point), m • P = 0 → (Affine.Point.map ↑σ) P = P

The m-division field is unramified exactly when inertia fixes E(K^{alg})[m]. For W elliptic, m invertible in K, and a structure of nonarchimedean local field on K(E[m]) compatible with K, the extension K(E[m])/K is unramified exactly when every element of the inertia subgroup of Gal(K^{alg}/K) fixes every point of E(K^{alg})[m].

theorem WeierstrassCurve.isUnramified_divisionField_pow_of_tateModuleGaloisRepresentation_eq_one {K : Type u_1} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] (W : WeierstrassCurve K) [W.IsElliptic] {ℓ : ℕ} [Fact (Nat.Prime ℓ)] (hℓ : ↑ℓ ≠ 0) (h : ∀ σ ∈ TauCeti.inertiaSubgroup K, (W.tateModuleGaloisRepresentation ℓ) σ = 1) (k : ℕ) [ValuativeRel ↥(W.divisionField (ℓ ^ k))] [TopologicalSpace ↥(W.divisionField (ℓ ^ k))] [IsNonarchimedeanLocalField ↥(W.divisionField (ℓ ^ k))] [ValuativeExtension K ↥(W.divisionField (ℓ ^ k))] : TauCeti.IsUnramified K ↥(W.divisionField (ℓ ^ k))

An unramified Tate module gives unramified division fields. Let ℓ be a prime invertible in K. If the inertia subgroup of Gal(K^{alg}/K) acts trivially on the ℓ-adic Tate module of an elliptic curve W, then for every k the ℓ ^ k-division field K(E[ℓ ^ k]) is unramified over K, for every structure of nonarchimedean local field on it compatible with K.