The n-torsion of an elliptic curve is étale exactly where n is invertible

Let E be an elliptic curve over a scheme S and n a nonzero integer. The n-torsion E[n] is finite locally free of rank n² over S. This file proves that it is étale over S exactly when n is invertible in every residue field of S (Katz–Mazur 2.3.1).

Where n is invertible, E[n] is flat and locally of finite presentation over S, and formally unramified (EllipticCurveGeom.formallyUnramified_torsion_hom), so it is étale. Over a residue field of characteristic p dividing n, an elliptic curve has fewer than n² points of order dividing n, even over an algebraically closed field: it has at most p points of order dividing p (Silverman III.6.4). A finite étale scheme over a separably closed field has as many rational points as its rank (AlgebraicGeometry.Scheme.Hom.natCard_eq_finrank_of_etale), so the base change of E[n] to an algebraic closure of that residue field, of rank n², is not étale, and neither is E[n].

Main results

• TauCeti.AlgebraicGeometry.EllipticCurveGeom.natCard_torsion_lt: over a field in which the nonzero integer n vanishes, E[n] has fewer than n² points with values in the field.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_torsion_hom_of_forall_intCast_ne_zero: if n is invertible in every residue field of S, then E[n] is étale over S; it is finite over S, so it is finite étale.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_subscheme_of_ker_torsionι_le: a flat, locally finitely presented closed subscheme of E contained in E[n] is étale over S where n is invertible.
• TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_torsion_hom_iff: for n ≠ 0, E[n] is étale over S if and only if n is invertible in every residue field of S.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, 2.3.1.
• J. H. Silverman, The Arithmetic of Elliptic Curves, III.6.4.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.natCard_torsion_lt {K : Type u} [Field K] (E : EllipticCurveGeom (AlgebraicGeometry.Spec ↧K)) {n : ℤ} (hn : n ≠ 0) (hnK : ↑n = 0) : Nat.card (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over (AlgebraicGeometry.Spec ↧K)) ⟶ (E.torsion n).X) < n.natAbs ^ 2

An elliptic curve has fewer than n² points of order dividing n when n vanishes in the field. For an elliptic curve E over a field K and a nonzero integer n with (n : K) = 0, the n-torsion E[n] has fewer than n.natAbs ^ 2 points with values in Spec K, although it has rank n.natAbs ^ 2 over Spec K.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_torsion_hom_of_forall_intCast_ne_zero {S : AlgebraicGeometry.Scheme} (E : EllipticCurveGeom S) {n : ℤ} (hn : ∀ (s : ↥S), ↑n ≠ 0) : AlgebraicGeometry.Etale (E.torsion n).X.hom

The n-torsion of an elliptic curve is étale where n is invertible (Katz–Mazur 2.3.1). For an elliptic curve E over a scheme S and an integer n invertible in every residue field of S, the n-torsion E[n] is étale over S. It is also finite over S (EllipticCurveGeom.isFinite_torsion_hom), so E[n] is finite étale over S.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_subscheme_of_ker_torsionι_le {S : AlgebraicGeometry.Scheme} (E : EllipticCurveGeom S) (I : E.carrier.IdealSheafData) {n : ℤ} [AlgebraicGeometry.Flat (CategoryTheory.CategoryStruct.comp I.subschemeι E.structureMap)] [AlgebraicGeometry.LocallyOfFinitePresentation (CategoryTheory.CategoryStruct.comp I.subschemeι E.structureMap)] (hn : ∀ (s : ↥S), ↑n ≠ 0) (hle : AlgebraicGeometry.Scheme.Hom.ker (CategoryTheory.Over.Hom.left (E.torsionι n)) ≤ I) : AlgebraicGeometry.Etale (CategoryTheory.CategoryStruct.comp I.subschemeι E.structureMap)

A flat, locally finitely presented closed subscheme of E contained in E[n] is étale over the base when n is invertible in every residue field. Containment is expressed as the reverse inclusion of ideal sheaves.

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.etale_torsion_hom_iff {S : AlgebraicGeometry.Scheme} (E : EllipticCurveGeom S) {n : ℤ} (hn : n ≠ 0) : AlgebraicGeometry.Etale (E.torsion n).X.hom ↔ ∀ (s : ↥S), ↑n ≠ 0

Criterion for E[n] to be étale (Katz–Mazur 2.3.1). For an elliptic curve E over a scheme S and a nonzero integer n, the n-torsion E[n] is étale over S if and only if n is invertible in every residue field of S.