Existence of globally minimal and sharp semi-global Weierstrass equations

Let O be a Dedekind domain with fraction field K, and E an elliptic curve over K. The global-minimality class globalMinimalityClass O E ∈ ClassGroup O is the class of the defect ideal 𝔍_W = ∏ᵥ 𝔭ᵥ ^ fᵥ(W) of any integral model W of E. This file proves that it is the complete obstruction: E has a globally minimal Weierstrass equation over O if and only if its global-minimality class is trivial (Silverman, AEC, Proposition VIII.8.2). In particular every elliptic curve over the fraction field of a principal ideal domain, such as ℚ, has a globally minimal equation (Corollary VIII.8.3).

When the class is not trivial, the same patching keeps the defect at a single prime: E has a sharp semi-global model at a height-one prime v₀ — integral at v₀, minimal at every other prime, with obstruction exponent exactly one at v₀ — if and only if its global-minimality class is the class of 𝔭_{v₀}.

Main results

• WeierstrassCurve.globalMinimalityClass_eq_one_iff: globalMinimalityClass O E = 1 if and only if some change of variables makes E globally minimal over O.
• WeierstrassCurve.exists_isGlobalMinimal_smul_of_weierstrassDefectIdeal_eq_span: an integral model with a principal defect ideal has a globally minimal model with the prescribed scaling factor.
• WeierstrassCurve.exists_isGlobalMinimal_smul: over the fraction field of a principal ideal domain every elliptic curve has a globally minimal equation.
• WeierstrassCurve.exists_isSharpSemiGlobalMinimalAt_smul_of_coe_weierstrassDefectIdeal_eq: an integral model whose defect ideal is (u) · 𝔭_{v₀} has a sharp semi-global model at v₀ with scaling factor u.
• WeierstrassCurve.globalMinimalityClass_eq_classGroupMk_iff: globalMinimalityClass O E is the class of 𝔭_{v₀} if and only if some change of variables makes E sharply semi-global at v₀.

The obstruction

A globally minimal equation has trivial defect ideal, so a trivial global-minimality class is necessary. Conversely, if an integral model W has principal defect ideal (g), then some change of variables with scaling factor g carries W to a globally minimal equation (exists_isGlobalMinimal_smul_of_weierstrassDefectIdeal_eq_span); this is the patching of local minimal models in Silverman's proof of VIII.8.2.

The sharp semi-global model is built by the same patching. If 𝔍_W = (u) · 𝔭_{v₀}, then u has order fᵥ(W) at every v ≠ v₀ and order f_{v₀}(W) - 1 at v₀. The translation parameters are chosen exactly as for a globally minimal model, so the resulting equation is minimal away from v₀; at v₀ it is the minimal local model rescaled by a change of variables whose u⁻¹ is a uniformiser up to a unit, which keeps it integral and raises the obstruction exponent from zero to one.

The argument uses nothing about O beyond finite approximation, so it is stated for an arbitrary Dedekind domain; the ring of integers of a number field is the case Silverman treats.

References

• J. Silverman, The Arithmetic of Elliptic Curves, VII.1.3 and VIII.8.2–8.3.

Comparing with a local change of variables at one prime

The local data at every prime

Patching

theorem WeierstrassCurve.exists_isGlobalMinimal_smul_of_weierstrassDefectIdeal_eq_span {O : Type u_1} [CommRing O] [IsDedekindDomain O] {K : Type u_2} [Field K] [Algebra O K] [IsFractionRing O K] (W : WeierstrassCurve K) [W.IsElliptic] [IsIntegral O W] {g : O} (hg : weierstrassDefectIdeal O W = Ideal.span {g}) : ∃ (C : VariableChange K), ↑C.u = (algebraMap O K) g ∧ IsGlobalMinimal O (C • W)

An integral model with principal defect ideal has a globally minimal model (the patching step of Silverman VIII.8.2): if g generates the defect ideal of the integral model W, then some change of variables with scaling factor g carries W to an equation minimal at every height-one prime of O.

Sharp semi-global models

theorem WeierstrassCurve.exists_isSharpSemiGlobalMinimalAt_smul_of_coe_weierstrassDefectIdeal_eq {O : Type u_1} [CommRing O] [IsDedekindDomain O] {K : Type u_2} [Field K] [Algebra O K] [IsFractionRing O K] (W : WeierstrassCurve K) [W.IsElliptic] [IsIntegral O W] (v₀ : IsDedekindDomain.HeightOneSpectrum O) {u : K} (hu : ↑(weierstrassDefectIdeal O W) = FractionalIdeal.spanSingleton (nonZeroDivisors O) u * ↑v₀.asIdeal) : ∃ (C : VariableChange K), ↑C.u = u ∧ IsSharpSemiGlobalMinimalAt O v₀ (C • W)

An integral model whose defect ideal is (u) · 𝔭_{v₀} has a sharp semi-global model at v₀ with scaling factor u. The change of variables is minimal away from v₀, where u has the order of the defect, and leaves an obstruction exponent of exactly one at v₀, where the order of u falls one short of it. This is the patching step of Silverman VIII.8.2 with one prime kept back.

The global-minimality class is the complete obstruction

theorem WeierstrassCurve.globalMinimalityClass_eq_one_iff (O : Type u_1) [CommRing O] [IsDedekindDomain O] {K : Type u_2} [Field K] [Algebra O K] [IsFractionRing O K] (E : WeierstrassCurve K) [E.IsElliptic] : globalMinimalityClass O E = 1 ↔ ∃ (C : VariableChange K), IsGlobalMinimal O (C • E)

The global-minimality class is the obstruction to a globally minimal equation (Silverman, AEC, Proposition VIII.8.2): an elliptic curve E over the fraction field K of a Dedekind domain O has a Weierstrass equation minimal at every height-one prime of O if and only if its global-minimality class in ClassGroup O is trivial.

theorem WeierstrassCurve.globalMinimalityClass_eq_classGroupMk_iff (O : Type u_1) [CommRing O] [IsDedekindDomain O] {K : Type u_2} [Field K] [Algebra O K] [IsFractionRing O K] (E : WeierstrassCurve K) [E.IsElliptic] (v₀ : IsDedekindDomain.HeightOneSpectrum O) : globalMinimalityClass O E = v₀.classGroupMk ↔ ∃ (C : VariableChange K), IsSharpSemiGlobalMinimalAt O v₀ (C • E)

The global-minimality class is the class of a height-one prime v₀ exactly when the curve has a sharp semi-global model at v₀: an equation integral at v₀, minimal at every other height-one prime, and with obstruction exponent one at v₀, so that its defect ideal is 𝔭_{v₀} itself. No condition on v₀ is needed, in particular none on its residue characteristic. Over the ring of integers of a number field every ideal class contains a prime, so every elliptic curve there has such a model at a suitable prime; that fact about class groups is not part of this statement.

theorem WeierstrassCurve.exists_isGlobalMinimal_smul (O : Type u_1) [CommRing O] [IsDedekindDomain O] {K : Type u_2} [Field K] [Algebra O K] [IsFractionRing O K] [IsPrincipalIdealRing O] (E : WeierstrassCurve K) [E.IsElliptic] : ∃ (C : VariableChange K), IsGlobalMinimal O (C • E)

Over the fraction field of a principal ideal domain every elliptic curve has a globally minimal Weierstrass equation (Silverman, AEC, Corollary VIII.8.3), the class group being trivial. This applies to ℚ over ℤ, and to a number field of class number one over its ring of integers.