Comparison of Drinfeld bases and naive full level structures

Let E be an elliptic curve over a scheme S and N a positive integer. This file compares Drinfeld bases of E[N] (EllipticCurveGeom.IsDrinfeldBasis) with naive [Γ(N)]-structures (EllipticCurveGeom.IsFiberwiseInjective), following Katz–Mazur §3.7:

1. Over any base scheme, every naive [Γ(N)]-structure is a Drinfeld basis (EllipticCurveGeom.IsFiberwiseInjective.isDrinfeldBasis). The section divisors [φ a] are pairwise disjoint, so their sum is their scheme-theoretic union; it is contained in E[N] and has the same rank N², so finite flat rank comparison identifies the two closed subschemes. No invertibility hypothesis on N is needed for this direction, and the argument works over nonreduced and empty bases.
2. Where N is invertible on the base, every Drinfeld basis is a naive [Γ(N)]-structure (EllipticCurveGeom.IsDrinfeldBasis.isFiberwiseInjective), so the two notions coincide (EllipticCurveGeom.isDrinfeldBasis_iff_isFiberwiseInjective). Over any field K, the divisor equality ∑_a [φ a] = E[N] forces every K-point of E[N] to lie on one of the sections φ a, so the image of φ is E[N](K) (EllipticCurveGeom.IsDrinfeldBasis.range_eq_ker). Over a separably closed field in which N is invertible, E[N](K) has N² points, so φ is injective (EllipticCurveGeom.isFiberwiseInjective_iff_range_eq_ker). 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.IsFiberwiseInjective.isDrinfeldBasis: every naive [Γ(N)]-structure is a Drinfeld basis, over an arbitrary base.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.left_apply_ne_zero: on a fibre where N is invertible, no φ a with a ≠ 1 meets the zero section.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.isFiberwiseInjective: where N is invertible on the base, every Drinfeld basis is a naive [Γ(N)]-structure.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff_isFiberwiseInjective: where N is invertible on the base, Drinfeld bases and naive [Γ(N)]-structures coincide.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff_injective: over a field in which N is invertible, a homomorphism (ℤ/Nℤ)² →* E(K) is a Drinfeld basis if and only if it is injective.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, §§1.4, 1.10, 3.1, 3.7.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.isDrinfeldBasis {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {N : ℕ} [NeZero N] {φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)} (hφ : E.IsFiberwiseInjective φ) : E.IsDrinfeldBasis N φ

Every naive full level structure is a Drinfeld basis, without requiring N to be invertible on the base.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.left_apply_ne_zero {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {N : ℕ} [NeZero N] {φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)} (h : E.IsDrinfeldBasis N φ) {s : ↥S} (hN : ↑N ≠ 0) {a : Multiplicative (ZMod N × ZMod N)} (ha : a ≠ 1) : (CategoryTheory.Over.Hom.left (φ a)) s ≠ E.zero s

A Drinfeld basis of E[N] is injective on every fibre where N is invertible. If φ is a Drinfeld basis of E[N] and N is invertible in the residue field at s, then for every a ≠ 1 in (ℤ/Nℤ)² the section φ a misses the zero section at s.

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

Every Drinfeld basis is a naive [Γ(N)]-structure where N is invertible (Katz–Mazur, §3.7): if N is invertible in every residue field of S, a Drinfeld basis φ : (ℤ/Nℤ)² →* E(S) is injective on every fibre of E over S.

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

Drinfeld bases and naive [Γ(N)]-structures coincide where N is invertible (Katz–Mazur, §3.7). If N is invertible in every residue field of S, a homomorphism φ : (ℤ/Nℤ)² →* E(S) is a Drinfeld basis of E[N] if and only if it is injective on every fibre of E over S.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsDrinfeldBasis.injective {N : ℕ} [NeZero N] {K : Type u} [Field K] {E : EllipticCurveGeom (AlgebraicGeometry.Spec ↧K)} {φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over (AlgebraicGeometry.Spec ↧K)) ⟶ CategoryTheory.Over.mk E.structureMap)} (h : E.IsDrinfeldBasis N φ) (hN : ↑N ≠ 0) : Function.Injective ⇑φ

Over a field in which N is invertible, a Drinfeld basis is injective.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff_injective {N : ℕ} [NeZero N] {K : Type u} [Field K] {E : EllipticCurveGeom (AlgebraicGeometry.Spec ↧K)} {φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over (AlgebraicGeometry.Spec ↧K)) ⟶ CategoryTheory.Over.mk E.structureMap)} (hN : ↑N ≠ 0) : E.IsDrinfeldBasis N φ ↔ Function.Injective ⇑φ

Over a field in which N is invertible, the Drinfeld bases of E[N] are the injective homomorphisms (ℤ/Nℤ)² →* E(K).