Being a Drinfeld basis is a closed condition

Let E be an elliptic curve over a scheme S, N a positive integer and φ : (ℤ/Nℤ)² →* E(S) a homomorphism to the group of S-points of E. Whether the base change of φ along a scheme S' over S is a Drinfeld basis of E'[N] is a closed condition on S': there is an ideal sheaf E.drinfeldBasisLocus N φ on S such that the base change of φ is a Drinfeld basis exactly when S' ⟶ S factors through its closed subscheme (EllipticCurveGeom.isDrinfeldBasis_pointsPullback_iff).

A Drinfeld basis is a homomorphism whose divisor D = ∑_{a ∈ (ℤ/Nℤ)²} [φ a] is the closed subscheme E[N]. Both D and E[N] are finite locally free over S and commute with base change, so the Drinfeld-basis locus is their equality locus (AlgebraicGeometry.Scheme.IdealSheafData.equalityLocus): the intersection of the locus where the finite locally free family D ⟶ E lies in E[N] and the locus where the finite locally free family E[N] ⟶ E lies in D, each cut out by a Weil restriction of a closed subscheme.

The locus commutes with base change (EllipticCurveGeom.drinfeldBasisLocus_pointsPullback), is invariant under automorphisms of (ℤ/Nℤ)², in particular under GL₂(ℤ/Nℤ) (EllipticCurveGeom.drinfeldBasisLocus_comp_mulEquiv), and over it the base change of φ is a Drinfeld basis (EllipticCurveGeom.isDrinfeldBasis_pointsPullback_baseChange_subschemeι). No hypothesis is placed on S, which may be nonreduced, and N need not be invertible on it: the condition is scheme-theoretic, not a condition on geometric fibres.

Applied to a universal homomorphism (ℤ/Nℤ)² →* E_H(H), of which every homomorphism (ℤ/Nℤ)² →* E_T(T) is the base change along a unique morphism T ⟶ H (for instance the one given by the pair of universal points over H = E[N] ×_S E[N]), the universal property says that the [Γ(N)]-structures on base changes of E are the points of the closed subscheme of H cut out by the Drinfeld-basis locus.

Main definitions

• TauCeti.AlgebraicGeometry.EllipticCurveGeom.drinfeldBasisLocus E N φ: the ideal sheaf on S of the locus over which φ is a Drinfeld basis of E[N].

Main results

• TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_pointsPullback_iff: the universal property of the Drinfeld-basis locus.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff_drinfeldBasisLocus_eq_bot: φ is a Drinfeld basis exactly when its locus is all of S.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.drinfeldBasisLocus_pointsPullback: the locus commutes with base change.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, 1.3.7 and 1.6.2 (conditions on finite locally free closed subschemes of a curve are closed), §1.10 (the closed subscheme of generators) and §3.1 ([Γ(N)]-structures).
• S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models, §7.6, for Weil restriction.

noncomputable def TauCeti.AlgebraicGeometry.EllipticCurveGeom.drinfeldBasisLocus {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)) : S.IdealSheafData

The ideal sheaf on S of the Drinfeld-basis locus of a homomorphism φ : (ℤ/Nℤ)² →* E(S) (Katz–Mazur, 1.6.2 and §3.1): a morphism S' ⟶ S factors through its closed subscheme exactly when the base change of φ is a Drinfeld basis of E'[N] (EllipticCurveGeom.isDrinfeldBasis_pointsPullback_iff). It is the equality locus of the divisor ∑_a [φ a] and E[N], both finite locally free over S.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_pointsPullback_iff {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {N : ℕ} [NeZero N] {S' : AlgebraicGeometry.Scheme} {E' : EllipticCurveGeom S'} {g : E'.carrier ⟶ E.carrier} {f : S' ⟶ S} (hg : CategoryTheory.IsPullback g E'.structureMap E.structureMap f) (h0 : CategoryTheory.CategoryStruct.comp E'.zero g = CategoryTheory.CategoryStruct.comp f E.zero) (φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)) : E'.IsDrinfeldBasis N ((pointsPullback hg h0).comp φ) ↔ E.drinfeldBasisLocus N φ ≤ AlgebraicGeometry.Scheme.Hom.ker f

The universal property of the Drinfeld-basis locus (Katz–Mazur, 1.6.2 and §3.1). Let g : E' ⟶ E be a morphism of elliptic curves lying over f : S' ⟶ S, forming a pullback square with the structure morphisms and carrying the zero section of E' to that of E. Then the base change (ℤ/Nℤ)² →* E'(S') of φ is a Drinfeld basis of E'[N] exactly when f factors through the closed subscheme of E.drinfeldBasisLocus N φ.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_iff_drinfeldBasisLocus_eq_bot {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)) : E.IsDrinfeldBasis N φ ↔ E.drinfeldBasisLocus N φ = ⊥

A homomorphism φ : (ℤ/Nℤ)² →* E(S) is a Drinfeld basis of E[N] exactly when its Drinfeld-basis locus is all of S.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.isDrinfeldBasis_pointsPullback_baseChange_subschemeι {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)) : (E.baseChange (E.drinfeldBasisLocus N φ).subschemeι).IsDrinfeldBasis N ((pointsPullback ⋯ ⋯).comp φ)

Over the Drinfeld-basis locus, φ is a Drinfeld basis: the base change of φ to the closed subscheme of E.drinfeldBasisLocus N φ is a Drinfeld basis.

@[simp]

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.drinfeldBasisLocus_comp_mulEquiv {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)) (σ : Multiplicative (ZMod N × ZMod N) ≃* Multiplicative (ZMod N × ZMod N)) : E.drinfeldBasisLocus N (φ.comp ↑σ) = E.drinfeldBasisLocus N φ

The Drinfeld-basis locus is invariant under automorphisms of (ℤ/Nℤ)², in particular under the right action of GL₂(ℤ/Nℤ): reindexing the points φ a does not change their divisor.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.drinfeldBasisLocus_pointsPullback {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {N : ℕ} [NeZero N] {S' : AlgebraicGeometry.Scheme} {E' : EllipticCurveGeom S'} {g : E'.carrier ⟶ E.carrier} {f : S' ⟶ S} (hg : CategoryTheory.IsPullback g E'.structureMap E.structureMap f) (h0 : CategoryTheory.CategoryStruct.comp E'.zero g = CategoryTheory.CategoryStruct.comp f E.zero) (φ : Multiplicative (ZMod N × ZMod N) →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)) : E'.drinfeldBasisLocus N ((pointsPullback hg h0).comp φ) = (E.drinfeldBasisLocus N φ).comap f

The Drinfeld-basis locus commutes with base change. For a morphism g : E' ⟶ E of elliptic curves forming a pullback square over f : S' ⟶ S and carrying the zero section of E' to that of E, the Drinfeld-basis locus of the base change of φ is the inverse image along f of the Drinfeld-basis locus of φ.