Points of exact order N where N is invertible

Let E be an elliptic curve over a scheme S and P an S-point of E of exact order N in the sense of Drinfeld (EllipticCurveGeom.IsOfExactOrder): the divisor D = [P] + [2P] + ⋯ + [NP] is a subgroup scheme of E. This file shows that if N is invertible in every residue field of S, then P also has exact order N in the naive sense: none of P, 2P, …, (N - 1)P meets the zero section, so P has order exactly N in every geometric fibre. This is one half of the comparison between Drinfeld and naive [Γ₁(N)]-structures where N is invertible. It fails in characteristic p dividing N, where the zero section can have exact order p in the sense of Drinfeld.

The subgroup scheme D lies in E[N] (EllipticCurveGeom.IsOfExactOrder.ker_torsionι_le), which is étale over S where N is invertible. Being flat and finitely presented, the closed subscheme D is then itself étale over S (EllipticCurveGeom.IsOfExactOrder.etale_multiplesDivisor). Over a separably closed field, D therefore has exactly N points, its rank. But every point of D is one of P, 2P, …, NP, since the support of D is the union of their images; so these N points are pairwise distinct, and P has order N. The general case reduces to this one by base change to the algebraic closures of the residue fields of S.

Main results

• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.etale_multiplesDivisor: where N is invertible, the divisor of a point of exact order N is finite étale over the base.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.pow_left_apply_ne_zero: on a fibre where N is invertible, none of P, 2P, …, (N - 1)P meets the zero section.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.orderOf_eq: over a field in which N is invertible, a point of exact order N has order N.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.isFiberwiseInjective: where N is invertible, the homomorphism ℤ/Nℤ →* E(S) of a point of exact order N is injective on every fibre, that is, a naive [Γ₁(N)]-structure.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §1.4 (points of exact order N) and §3.7 (comparison with naive level structures where N is invertible).

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.etale_multiplesDivisor {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {P : CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap} {N : ℕ} (h : E.IsOfExactOrder P N) (hN : ∀ (s : ↥S), ↑N ≠ 0) : AlgebraicGeometry.Etale (CategoryTheory.CategoryStruct.comp (E.multiplesDivisor P N).subschemeι E.structureMap)

The subgroup divisor of a point of exact order N is étale where N is invertible. If P has exact order N and N is invertible in every residue field of S, then the divisor [P] + ⋯ + [NP] is étale over S. It is also finite over S, so it is finite étale.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.pow_left_apply_ne_zero {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {P : CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap} {N : ℕ} (h : E.IsOfExactOrder P N) {s : ↥S} (hN : ↑N ≠ 0) {k : ℕ} (hk : 0 < k) (hkN : k < N) : (CategoryTheory.Over.Hom.left (P ^ k)) s ≠ E.zero s

A point of exact order N has exact order N on every fibre where N is invertible. If P has exact order N in the sense of Drinfeld and N is invertible in the residue field at s, then none of P, 2P, …, (N - 1)P meets the zero section at s.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.isFiberwiseInjective {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {N : ℕ} [NeZero N] {φ : Multiplicative (ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)} (h : E.IsOfExactOrder (φ (Multiplicative.ofAdd 1)) N) (hN : ∀ (s : ↥S), ↑N ≠ 0) : E.IsFiberwiseInjective φ

A point of exact order N gives a naive [Γ₁(N)]-structure where N is invertible. If N is invertible in every residue field of S and φ : ℤ/Nℤ →* E(S) sends 1 to a point of exact order N, then φ is injective on every fibre of E over S.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsOfExactOrder.orderOf_eq {K : Type u} [Field K] {E : EllipticCurveGeom (AlgebraicGeometry.Spec ↧K)} {P : CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over (AlgebraicGeometry.Spec ↧K)) ⟶ CategoryTheory.Over.mk E.structureMap} {N : ℕ} (h : E.IsOfExactOrder P N) (hN : ↑N ≠ 0) : orderOf P = N

Over a field in which N is invertible, a point of exact order N has order N.