The ℓ-adic Galois representation of an elliptic curve

Let W be a Weierstrass curve over a field F, let K be an extension of F, and let ℓ be a prime. An F-automorphism σ of K acts on the points of W over K coordinatewise, by group automorphisms, so it permutes each torsion level E[ℓ ^ n] compatibly with the transition maps. It therefore acts ℤ_ℓ-linearly on the Tate module T_ℓ E = lim E[ℓ ^ n], and this file packages that action as a representation of Gal(K/F) on T_ℓ E (Silverman III.7).

For an elliptic curve the representation is continuous for the Krull topology on Gal(K/F) and the inverse-limit topology on T_ℓ E, with no hypothesis on K / F. The coordinates of a torsion point are integral over F — the abscissa is a root of a division polynomial and the ordinate then solves the Weierstrass equation — so an automorphism near σ moves each torsion point as σ does. When K is a separable closure of F and ℓ is invertible in F, the Tate module is free of rank two (WeierstrassCurve.finrank_tateModule), so this is the classical two-dimensional ℓ-adic representation; its matrices in GL₂(ℤ_ℓ) depend on a choice of basis, so it is stated basis-free, on T_ℓ E itself.

Main definitions

• WeierstrassCurve.tateModuleGaloisRepresentation: the representation of K ≃ₐ[F] K on T_ℓ E.

Main results

• WeierstrassCurve.coe_proj_tateModuleGaloisRepresentation: on the level E[ℓ ^ n] the representation is the coordinatewise action on points.
• WeierstrassCurve.isLocallyConstant_map_of_zsmul_eq_zero: the orbit map of a torsion point of an elliptic curve is locally constant for the Krull topology.
• WeierstrassCurve.continuous_tateModuleGaloisRepresentation_apply: for an elliptic curve, the action (σ, x) ↦ σ x on T_ℓ E is jointly continuous.

References

• J. Silverman, The Arithmetic of Elliptic Curves, III.7.

The representation

noncomputable def WeierstrassCurve.tateModuleGaloisRepresentation {F : Type u_1} {K : Type u_2} [Field F] [Field K] [DecidableEq K] [Algebra F K] (W : WeierstrassCurve F) (ℓ : ℕ) [Fact (Nat.Prime ℓ)] : Representation ℤ_[ℓ] Gal(K/F) (TauCeti.TateModule ℓ (W.baseChange K).toAffine.Point)

The ℓ-adic Galois representation of W: an F-automorphism σ of K acts on the Tate module T_ℓ E of W over K by applying σ to the coordinates of every point of every level E[ℓ ^ n].

Equations

• W.tateModuleGaloisRepresentation ℓ = { toFun := fun (σ : Gal(K/F)) => TauCeti.TateModule.mapLinearMap (WeierstrassCurve.Affine.Point.map ↑σ), map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem WeierstrassCurve.tateModuleGaloisRepresentation_apply {F : Type u_1} {K : Type u_2} [Field F] [Field K] [DecidableEq K] [Algebra F K] {W : WeierstrassCurve F} {ℓ : ℕ} [Fact (Nat.Prime ℓ)] (σ : Gal(K/F)) (x : TauCeti.TateModule ℓ (W.baseChange K).toAffine.Point) : ((W.tateModuleGaloisRepresentation ℓ) σ) x = (TauCeti.TateModule.map (Affine.Point.map ↑σ)) x

The representation applies the induced map of Tate modules of the coordinatewise action on points.

theorem WeierstrassCurve.coe_proj_tateModuleGaloisRepresentation {F : Type u_1} {K : Type u_2} [Field F] [Field K] [DecidableEq K] [Algebra F K] {W : WeierstrassCurve F} {ℓ : ℕ} [Fact (Nat.Prime ℓ)] (σ : Gal(K/F)) (x : TauCeti.TateModule ℓ (W.baseChange K).toAffine.Point) (n : ℕ) : ↑((TauCeti.TateModule.proj n) (((W.tateModuleGaloisRepresentation ℓ) σ) x)) = (Affine.Point.map ↑σ) ↑((TauCeti.TateModule.proj n) x)

The representation on the level E[ℓ ^ n]: the n-th component of σ x is σ applied to the coordinates of the n-th component of x.

Continuity

theorem WeierstrassCurve.isLocallyConstant_map_of_zsmul_eq_zero {F : Type u_1} {K : Type u_2} [Field F] [Field K] [DecidableEq K] [Algebra F K] (W : WeierstrassCurve F) [W.IsElliptic] {n : ℤ} (hn : n ≠ 0) {P : (toAffine (Affine.baseChange W K)).Point} (hP : n • P = 0) : IsLocallyConstant fun (σ : Gal(K/F)) => (Affine.Point.map ↑σ) P

Torsion points move locally constantly under the Galois group. For a point P of an elliptic curve over K killed by a nonzero integer, σ ↦ σ P is locally constant for the Krull topology on K ≃ₐ[F] K: the coordinates of P are integral over F, so they are fixed by an open subgroup. No hypothesis on K / F is needed.

theorem WeierstrassCurve.continuous_tateModuleGaloisRepresentation_apply {F : Type u_1} {K : Type u_2} [Field F] [Field K] [DecidableEq K] [Algebra F K] (W : WeierstrassCurve F) [W.IsElliptic] (ℓ : ℕ) [Fact (Nat.Prime ℓ)] : Continuous fun (q : Gal(K/F) × TauCeti.TateModule ℓ (W.baseChange K).toAffine.Point) => ((W.tateModuleGaloisRepresentation ℓ) q.1) q.2

The ℓ-adic Galois representation of an elliptic curve is continuous: the action (σ, x) ↦ σ x of K ≃ₐ[F] K with its Krull topology on T_ℓ E with its inverse-limit topology is jointly continuous.