Supersingularity and the Verschiebung

Let W be an elliptic curve over a field K of exponential characteristic p. This file proves that W is supersingular — it has no nonzero geometric point of order p — exactly when multiplication by p is purely inseparable (has separable degree 1), and exactly when the Verschiebung V : W⁽ᵖ⁾ → W is purely inseparable (Silverman V.3.1). The definition of supersingularity reads the p-torsion over an algebraic closure; the criterion reads only the function fields over K, so it makes no choice of closure and holds verbatim over imperfect and transcendental base fields.

Both implications go through the separable degree of [p].

• If the geometric p-torsion is trivial, [p] is purely inseparable. Two K-embeddings of K(W) into an algebraic closure Ω of K(W) that agree on [p]^* K(W) move the generic point by a p-torsion point of W(Ω) (TauCeti.Isogeny.zsmul_map_sub_map_genericPoint_eq_zero). That point is zero, and an embedding is determined by the image of the generic point, so there is only one embedding: the separable degree of [p] is 1.
• If V is purely inseparable, the geometric p-torsion is trivial. Base change along K → Kᵃˡᵍ carries V ∘ F_{W/K} = [p] to a factorisation of [p] of W over Kᵃˡᵍ. The base change of relative Frobenius is purely inseparable of degree p, so that of V has degree p as well; it is not separable, since separability is invariant under base change, so it is purely inseparable. Hence [p] over Kᵃˡᵍ has separable degree 1, so it has no kernel, and its kernel is the geometric p-torsion.

Since [p] = V ∘ F_{W/K} with F_{W/K} purely inseparable, [p] and V have the same separable degree, which connects the two.

The same comparison bounds the p-torsion of W over K itself: the kernel of [p] has at most as many points as its separable degree, which is at most deg V = p. So E[p] is either zero or cyclic of order p, and it is the latter over an algebraically closed field exactly when W is ordinary. More generally, when p divides n ≠ 0, the n-torsion of W over K has fewer than n² points, although [n] has degree n². Over a separably closed field a finite étale n-torsion scheme of rank n² would have n² points, so this bound shows that the n-torsion is not étale where the characteristic divides n.

Main results

• WeierstrassCurve.isSupersingular_iff_separableDegree_mulByIntIsogenyOfNeZero_eq_one and WeierstrassCurve.isSupersingular_iff_separableDegree_mulByIntIsogenyOfNeZero_pow_eq_one: W is supersingular exactly when [p], or any [p ^ k] with k ≠ 0, has separable degree 1.
• WeierstrassCurve.isSupersingular_iff_isPurelyInseparable_verschiebungIsogeny: W is supersingular exactly when its Verschiebung is purely inseparable.
• WeierstrassCurve.isOrdinary_iff_isSeparable_verschiebungIsogeny: in characteristic p > 0, W is ordinary exactly when its Verschiebung is separable.
• WeierstrassCurve.isOrdinary_iff_separableDegree_mulByIntIsogenyOfNeZero_eq and WeierstrassCurve.isOrdinary_iff_separableDegree_mulByIntIsogenyOfNeZero_pow_eq: in characteristic p > 0, W is ordinary exactly when [p ^ k] has separable degree p ^ k, for k = 1 or any k ≠ 0.
• WeierstrassCurve.natCard_torsionBy_le and WeierstrassCurve.torsionBy_eq_zmultiples: W has at most p points of order dividing p, so a nonzero one generates them all.
• WeierstrassCurve.natCard_torsionBy_lt: in characteristic p > 0, when p ∣ n ≠ 0, W has fewer than n² points of order dividing n, which detects the failure of the n-torsion to be étale.

References

• J. Silverman, The Arithmetic of Elliptic Curves, III.6.1, III.6.4 and V.3.1.

theorem WeierstrassCurve.isSupersingular_iff_separableDegree_mulByIntIsogenyOfNeZero_eq_one {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] : IsSupersingular p W ↔ (TauCeti.Isogeny.mulByIntIsogenyOfNeZero W.toAffine ⋯).separableDegree = 1

Supersingularity is pure inseparability of [p]: W is supersingular exactly when multiplication by p has separable degree 1 (Silverman V.3.1).

theorem WeierstrassCurve.isSupersingular_iff_separableDegree_mulByIntIsogenyOfNeZero_pow_eq_one {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] {k : ℕ} (hk : k ≠ 0) : IsSupersingular p W ↔ (TauCeti.Isogeny.mulByIntIsogenyOfNeZero W.toAffine ⋯).separableDegree = 1

Supersingularity is pure inseparability of [p ^ k]: for k ≠ 0, W is supersingular exactly when multiplication by p ^ k has separable degree 1 (Silverman V.3.1).

theorem WeierstrassCurve.isSupersingular_iff_isPurelyInseparable_verschiebungIsogeny {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] : IsSupersingular p W ↔ IsPurelyInseparable (↥(TauCeti.Isogeny.verschiebungIsogeny p W.toAffine).fieldPullback.fieldRange) (W.map (frobenius K p)).toAffine.FunctionField

Supersingularity is pure inseparability of the Verschiebung: W is supersingular exactly when V : W⁽ᵖ⁾ → W, the dual of relative Frobenius, is purely inseparable (Silverman V.3.1).

theorem WeierstrassCurve.isOrdinary_iff_isSeparable_verschiebungIsogeny {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] (hp : Nat.Prime p) : IsOrdinary p W ↔ Algebra.IsSeparable (↥(TauCeti.Isogeny.verschiebungIsogeny p W.toAffine).fieldPullback.fieldRange) (W.map (frobenius K p)).toAffine.FunctionField

Ordinarity is separability of the Verschiebung: in characteristic p > 0, W is ordinary exactly when V : W⁽ᵖ⁾ → W is separable. The Verschiebung has prime degree p, so it is separable or purely inseparable and not both. In characteristic zero (p = 1) every curve is supersingular in this sense and V is an isomorphism, so the statement needs p prime.

theorem WeierstrassCurve.isOrdinary_iff_separableDegree_mulByIntIsogenyOfNeZero_eq {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] (hp : Nat.Prime p) : IsOrdinary p W ↔ (TauCeti.Isogeny.mulByIntIsogenyOfNeZero W.toAffine ⋯).separableDegree = p

Ordinarity in terms of the separable degree of [p]: in characteristic p > 0, W is ordinary exactly when multiplication by p has separable degree p, the degree of the Verschiebung (Silverman V.3.1).

theorem WeierstrassCurve.isOrdinary_iff_separableDegree_mulByIntIsogenyOfNeZero_pow_eq {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] (hp : Nat.Prime p) {k : ℕ} (hk : k ≠ 0) : IsOrdinary p W ↔ (TauCeti.Isogeny.mulByIntIsogenyOfNeZero W.toAffine ⋯).separableDegree = p ^ k

Ordinarity in terms of the separable degree of [p ^ k]: in characteristic p > 0 and for k ≠ 0, W is ordinary exactly when multiplication by p ^ k has separable degree p ^ k (Silverman V.3.1).

theorem WeierstrassCurve.natCard_torsionBy_le {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] : Nat.card ↥(AddSubgroup.torsionBy W.toAffine.Point ↑p) ≤ p

An elliptic curve has at most p points of order dividing p in characteristic p (Silverman III.6.4(c)): the p-torsion is the kernel of [p], which has at most as many points as the separable degree of [p], the separable degree of the Verschiebung, of degree p. So E[p] is zero or cyclic of order p.

theorem WeierstrassCurve.torsionBy_eq_zmultiples {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] {Q : W.toAffine.Point} (hQ : p • Q = 0) (hQ₀ : Q ≠ 0) : AddSubgroup.torsionBy W.toAffine.Point ↑p = AddSubgroup.zmultiples Q

A nonzero point of order dividing p generates the p-torsion in characteristic p: the p-torsion has at most p points, and the point has order exactly p.

theorem WeierstrassCurve.natCard_torsionBy_lt {K : Type u_1} [Field K] (p : ℕ) [ExpChar K p] (W : WeierstrassCurve K) [W.IsElliptic] (hp : Nat.Prime p) {n : ℤ} (hn : n ≠ 0) (hpn : ↑p ∣ n) : Nat.card ↥(AddSubgroup.torsionBy W.toAffine.Point n) < n.natAbs ^ 2

A curve has fewer than n² points of order dividing n when the characteristic p > 0 divides n ≠ 0, although [n] has degree n². This is the input that detects the failure of the n-torsion of W to be étale where the characteristic divides n: over a separably closed field, a finite étale n-torsion of rank n² would have exactly n² points.