Automorphisms of an elliptic curve fixing a full level structure

Let E be an elliptic curve over a scheme S and let σ : E ⟶ E be an automorphism of E over S fixing the zero section, given as a cartesian square over the identity of S. Let φ : A →* E(S) be a homomorphism injective on every fibre (EllipticCurveGeom.IsFiberwiseInjective), and let a, b ∈ A with a² ≠ 1, b ≠ 1 and b ≠ a^{±1}, so that P = φ a and Q = φ b satisfy P ≠ -P and Q ∉ {0, P, -P} in every fibre. If σ fixes P and Q, then σ is the identity (EllipticCurveGeom.IsFiberwiseInjective.eq_id_of_isPullback). For a naive full level N structure φ : (ℤ/Nℤ)² →* E(S) with N ≥ 3, the two basis vectors qualify, which gives the rigidity of the naive moduli problem [Γ(N)] (Katz–Mazur 2.7.2). No hypothesis is made on S: it may be nonreduced or non-affine. (A naive [Γ(N)]-structure forces N to be invertible on S, since (ℤ/pℤ)² does not embed into the p-torsion of an elliptic curve in characteristic p.)

The bound N ≥ 3 is sharp, and both sections are needed. For N ≤ 2, the automorphism [-1] fixes every section killed by N. On Y² + Y = X³ over a field containing a primitive cube root of unity ζ, the automorphism (X, Y) ↦ (ζX, Y) fixes P = (0, 0) and -P = (0, -1), and P ≠ -P.

The projective model. Over Spec R, an automorphism of projModel W over Spec R fixing the zero section is a change of variables C = (u, r, s, t) with C • W = W (WeierstrassCurve.existsUnique_eq_eqToHom_comp_projModelVariableChangeIso_hom). It commutes with negation, so if it fixes (x₁, y₁) and (x₂, y₂) it also fixes (x₁, -y₁ - a₁x₁ - a₃). When x₁ - x₂ and 2y₁ + a₁x₁ + a₃ are units, these three fixed points force C = 1 (WeierstrassCurve.VariableChange.eq_one_of_isUnit_sub), so the automorphism is the identity (WeierstrassCurve.eq_id_of_projModelAffinePoint_comp_eq).

Over a scheme. On a pointed Weierstrass chart c of E, a section P of E induces a section of the projective model of the equation of c (PointedWeierstrassChart.modelPoint), and σ induces an automorphism of that model (PointedWeierstrassChart.modelMap). If P misses the zero section, its model point is an affine point (x, y). The two unit conditions are checked modulo the maximal ideals of the coefficient ring of the chart: over a field, two points with the same x-coordinate are equal or opposite, and the fibrewise conditions on P and Q exclude both.

Main results

• TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.eq_id_of_isPullback: an automorphism of E over S fixing the zero section, φ a and φ b is the identity.

References

• N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, 2.7.2 and Chapter 4.
• J. H. Silverman, The Arithmetic of Elliptic Curves, III.1 (the change of variables).

Rigidity

theorem TauCeti.AlgebraicGeometry.EllipticCurveGeom.IsFiberwiseInjective.eq_id_of_isPullback {S : AlgebraicGeometry.Scheme} {E : EllipticCurveGeom S} {A : Type u_1} [Group A] {φ : A →* (CategoryTheory.MonoidalCategoryStruct.tensorUnit (CategoryTheory.Over S) ⟶ CategoryTheory.Over.mk E.structureMap)} (hφ : E.IsFiberwiseInjective φ) {a b : A} (ha : a * a ≠ 1) (hb : b ≠ 1) (hba : b ≠ a) (hba' : b ≠ a⁻¹) {σ : E.carrier ⟶ E.carrier} (hσ : CategoryTheory.IsPullback σ E.structureMap E.structureMap (CategoryTheory.CategoryStruct.id S)) (h0 : CategoryTheory.CategoryStruct.comp E.zero σ = E.zero) (hσa : CategoryTheory.CategoryStruct.comp (CategoryTheory.Over.Hom.left (φ a)) σ = CategoryTheory.Over.Hom.left (φ a)) (hσb : CategoryTheory.CategoryStruct.comp (CategoryTheory.Over.Hom.left (φ b)) σ = CategoryTheory.Over.Hom.left (φ b)) : σ = CategoryTheory.CategoryStruct.id E.carrier

Automorphisms fixing a full level structure are trivial. Let φ : A →* E(S) be injective on every fibre, and let a, b ∈ A with a² ≠ 1, b ≠ 1, b ≠ a and b ≠ a⁻¹. An endomorphism σ of E over S, cartesian over the identity of S (so an automorphism), which fixes the zero section, φ a and φ b is the identity. For a naive [Γ(N)]-structure (ℤ/Nℤ)² →* E(S) with N ≥ 3, the two basis vectors are such a and b.