| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace BogomolovPop |
|
|
| def proEllKernel (ell : ℕ) (G : Type*) [Group G] [TopologicalSpace G] : Subgroup G := |
| ⨅ (H : OpenNormalSubgroup G) (_ : IsPGroup ell (G ⧸ H.toSubgroup)), H.toSubgroup |
| |
| instance proEllKernel_normal (ell : ℕ) (G : Type*) [Group G] [TopologicalSpace G] : |
| (proEllKernel ell G).Normal := by |
| unfold proEllKernel |
| exact Subgroup.normal_iInf_normal fun H => |
| Subgroup.normal_iInf_normal fun _ => H.isNormal' |
| |
| abbrev ProEll (ell : ℕ) (F : Type*) [Field F] := |
| Field.absoluteGaloisGroup F ⧸ proEllKernel ell (Field.absoluteGaloisGroup F) |
| |
| def closedGammaThree (G : Type*) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] : |
| Subgroup G := |
| (⁅(commutator G).topologicalClosure, (⊤ : Subgroup G)⁆).topologicalClosure |
| |
| instance closedGammaThree_normal (G : Type*) [Group G] [TopologicalSpace G] |
| [IsTopologicalGroup G] : (closedGammaThree G).Normal := by |
| unfold closedGammaThree |
| exact Subgroup.is_normal_topologicalClosure _ |
| |
| abbrev PiA (ell : ℕ) (F : Type*) [Field F] := TopologicalAbelianization (ProEll ell F) |
| |
| abbrev PiC (ell : ℕ) (F : Type*) [Field F] := ProEll ell F ⧸ closedGammaThree (ProEll ell F) |
| |
| theorem closedGammaThree_le (G : Type*) [Group G] [TopologicalSpace G] |
| [IsTopologicalGroup G] : closedGammaThree G ≤ (commutator G).topologicalClosure := by |
| exact Subgroup.topologicalClosure_minimal _ (Subgroup.commutator_le_left _ _) |
| (Subgroup.isClosed_topologicalClosure _) |
| |
| def centralProjection (ell : ℕ) (F : Type*) [Field F] : PiC ell F →* PiA ell F := |
| QuotientGroup.map _ _ (MonoidHom.id _) (closedGammaThree_le _) |
| |
| abbrev Delta (ell : ℕ) (F : Type*) [Field F] := (centralProjection ell F).ker |
| |
| def sourceCommutator {G : Type*} [Group G] (x y : G) : G := x⁻¹ * y⁻¹ * x * y |
| |
| def abelianLift (ell : ℕ) (F : Type*) [Field F] (x : PiA ell F) : ProEll ell F := |
| Quotient.out x |
| |
| theorem abelianLift_spec (ell : ℕ) (F : Type*) [Field F] (x : PiA ell F) : |
| QuotientGroup.mk' _ (abelianLift ell F x) = x := Quotient.out_eq x |
| |
| def bracket (ell : ℕ) (F : Type*) [Field F] (x y : PiA ell F) : Delta ell F := |
| ⟨QuotientGroup.mk' _ (sourceCommutator (abelianLift ell F x) (abelianLift ell F y)), by |
| change centralProjection ell F _ = 1 |
| simp only [centralProjection, sourceCommutator, map_mul, map_inv] |
| simp [mul_comm]⟩ |
| |
| def BracketCompatible {ell : ℕ} {K L : Type*} [Field K] [Field L] |
| (φ : PiA ell L ≃ₜ* PiA ell K) : Prop := |
| ∃ ψ : Delta ell L ≃ₜ* Delta ell K, |
| ∀ σ τ, ψ (bracket ell L σ τ) = bracket ell K (φ σ) (φ τ) |
| |
| abbrev PerfectFieldClosure (F : Type*) [Field F] := perfectClosure F (AlgebraicClosure F) |
| |
| def PerfectIsom (k K l L : Type*) [Field k] [Field K] [Field l] [Field L] |
| [Algebra k K] [Algebra l L] := |
| { α : PerfectFieldClosure K ≃+* PerfectFieldClosure L // |
| α '' Set.range (algebraMap k (PerfectFieldClosure K)) = |
| Set.range (algebraMap l (PerfectFieldClosure L)) } |
| |
| def FrobeniusStep {k K l L : Type*} [Field k] [Field K] [Field l] [Field L] |
| [Algebra k K] [Algebra l L] (α β : PerfectIsom k K l L) : Prop := |
| ∃ p : ℕ, p.Prime ∧ (p : L) = 0 ∧ |
| ∀ x : PerfectFieldClosure K, β.1 x = (α.1 x) ^ p |
| |
| abbrev IsomF (k K l L : Type*) [Field k] [Field K] [Field l] [Field L] |
| [Algebra k K] [Algebra l L] := |
| Quotient (Relation.EqvGen.setoid (FrobeniusStep (k := k) (K := K) (l := l) (L := L))) |
| |
| abbrev IsomC (ell : ℕ) (K L : Type*) [Field K] [Field L] := |
| {φ : PiA ell L ≃ₜ* PiA ell K // BracketCompatible φ} |
| |
| def ScalarStep {ell : ℕ} [Fact ell.Prime] {K L : Type*} [Field K] [Field L] |
| (φ χ : IsomC ell K L) : Prop := |
| ∃ u : (PadicInt ell)ˣ, ∀ x : PiA ell L, |
| Filter.Tendsto (fun n : ℕ => (φ.1 x) ^ PadicInt.appr (u : PadicInt ell) n) |
| Filter.atTop (nhds (χ.1 x)) |
| |
| abbrev IsomCUnits (ell : ℕ) [Fact ell.Prime] (K L : Type*) [Field K] [Field L] := |
| Quotient (Relation.EqvGen.setoid (ScalarStep (ell := ell) (K := K) (L := L))) |
| |
| def absoluteToPiA (ell : ℕ) (F : Type*) [Field F] : |
| Field.absoluteGaloisGroup F →* PiA ell F := |
| (QuotientGroup.mk' _).comp (QuotientGroup.mk' _) |
| |
| def absoluteAction {F : Type*} [Field F] (σ : Field.absoluteGaloisGroup F) |
| (x : AlgebraicClosure F) : AlgebraicClosure F := by |
| change AlgebraicClosure F ≃ₐ[F] AlgebraicClosure F at σ |
| exact σ x |
| |
| def PullbackGraph {ell : ℕ} {k K l L : Type*} |
| [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L] |
| (α : PerfectIsom k K l L) (φ : IsomC ell K L) : Prop := |
| ∃ β : AlgebraicClosure K ≃+* AlgebraicClosure L, |
| (∀ x : PerfectFieldClosure K, β x = ((α.1 x : PerfectFieldClosure L) : |
| AlgebraicClosure L)) ∧ |
| ∀ σ : Field.absoluteGaloisGroup L, ∃ τ : Field.absoluteGaloisGroup K, |
| (∀ x : AlgebraicClosure K, |
| β (absoluteAction τ x) = absoluteAction σ (β x)) ∧ |
| φ.1 (absoluteToPiA ell L σ) = absoluteToPiA ell K τ |
| |
| def PhiGraph {ell : ℕ} [Fact ell.Prime] {k K l L : Type*} |
| [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L] |
| (a : IsomF k K l L) (b : IsomCUnits ell K L) : Prop := |
| ∃ α : PerfectIsom k K l L, ∃ φ : IsomC ell K L, |
| Quotient.mk _ α = a ∧ Quotient.mk _ φ = b ∧ PullbackGraph α φ |
| |
| theorem main_graph_injective (ell : ℕ) [Fact ell.Prime] |
| (k K l L : Type*) [Field k] [Field K] [Field l] [Field L] |
| [IsAlgClosed k] [IsAlgClosed l] [Algebra k K] [Algebra l L] |
| [Algebra.EssFiniteType k K] [Algebra.EssFiniteType l L] |
| (hk : (ell : k) ≠ 0) (hl : (ell : l) ≠ 0) |
| (hK : 2 ≤ Algebra.trdeg k K) (hL : 2 ≤ Algebra.trdeg l L) : |
| ∀ (a a' : IsomF k K l L) (b : IsomCUnits ell K L), |
| PhiGraph a b → PhiGraph a' b → a = a' := by |
| sorry |
| |
| end BogomolovPop |
| |
| end |
| |
| end OAI |
| |
