openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Arithmetic /ClassFieldRealization.lean
| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ClassFieldModulus | |
| namespace OAI | |
| /-! | |
| # Unconditional even ray realization of cyclic ideal characters | |
| The conductor of a totally real cyclic extension has no infinite part. | |
| Arithmetic Artin reciprocity and the common rational congruence modulus | |
| therefore supply the kernel law for every actual Frobenius ideal | |
| character. This gives their realization as even ray characters. | |
| -/ | |
| noncomputable section | |
| open scoped NumberField Classical | |
| namespace DukePrimeDegree | |
| open GlobalClassFieldTheory GlobalClassFieldTheory.GlobalClassFields | |
| open PrimeDegreePackets PrimeDegreePackets.GaloisIdealCharacter | |
| variable {K L : Type} [Field K] [NumberField K] [Field L] [NumberField L] | |
| variable [Algebra K L] [FiniteDimensional K L] [IsAbelianGalois K L] | |
| theorem idealCharacter_kernelLaw_of_even_modulus | |
| (m : RayClass.Modulus K) (hm : m.infinitePart = ∅) | |
| (hnorm : m.congruenceSubgroup ≤ (ideleClassNorm K L).range) | |
| (hunram : ∀ v : IsDedekindDomain.HeightOneSpectrum (𝓞 K), | |
| v ∉ m.finitePart.support → | |
| Algebra.IsUnramifiedAt (𝓞 K) (primeAbove K L v).val) | |
| (N : ℕ) | |
| (hN : ∀ v ∈ m.finitePart.support, | |
| algebraMap (𝓞 K) (v.adicCompletionIntegers K) (N : 𝓞 K) ∈ | |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ m.finitePart v) | |
| (χ : (L ≃ₐ[K] L) →* ℂˣ) : PrincipalRay.KernelLaw (idealHom K L χ) N := by | |
| intro a ha hcong | |
| let x : Kˣ := Units.mk0 (a : K) (by exact_mod_cast ha) | |
| have hx : IdeleGroup.principalIdele K x ∈ RayClass.idelePrimeToModulusSubgroup m := | |
| principalIdele_mem_even_ray_of_congruence m hm N hN a ha hcong | |
| let y : RayClass.idelePrimeToModulusSubgroup m := ⟨IdeleGroup.principalIdele K x, hx⟩ | |
| let I := RayClass.primeToIdealMap m y | |
| have hI : I ∈ RayClass.principalRayIdealSubgroup m := by | |
| refine ⟨y, ?_, rfl⟩ | |
| exact ⟨x, rfl⟩ | |
| have H := idealCharacter_principalRay_eq_one m hnorm hunram χ I hI | |
| have hval : (I : FractionalIdealGroup K) = integralIdealFractionalUnit | |
| (Ideal.span {a}) (by simpa only [ne_eq, Ideal.span_singleton_eq_bot] using ha) := by | |
| change IdeleGroup.fractionalIdeal (IdeleGroup.principalIdele K x) = _ | |
| rw [IdeleGroup.fractionalIdeal_principalIdele] | |
| apply Units.ext | |
| rw [coe_toPrincipalIdeal] | |
| exact (FractionalIdeal.coeIdeal_span_singleton a).symm | |
| change idealHom K L χ (I : FractionalIdealGroup K) = 1 at H | |
| change idealHom K L χ (integralIdealFractionalUnit (Ideal.span {a}) _) = 1 | |
| rwa [hval] at H | |
| end DukePrimeDegree | |
| namespace DukePrimeDegree | |
| open PrimeDegreePackets PrimeDegreePackets.GaloisIdealCharacter | |
| /-- Actual cyclic Frobenius characters of a totally real extension have | |
| one common even ray modulus, with the discriminant primes removed. -/ | |
| theorem exists_cyclic_ray_family_unconditional | |
| (K L : Type) [Field K] [NumberField K] [Field L] [NumberField L] | |
| [Algebra K L] [FiniteDimensional K L] [IsGalois K L] | |
| [IsCyclic (L ≃ₐ[K] L)] [NumberField.IsTotallyReal K] [NumberField.IsTotallyReal L] : | |
| ∃ N : ℕ, 0 < N ∧ (NumberField.discr L).natAbs ∣ N ∧ | |
| ∃ ψ : ((L ≃ₐ[K] L) →* ℂˣ) → EvenRayCharacter K N, | |
| (∀ χ, (ψ χ).idealHom = idealHom K L χ) ∧ | |
| ∀ χ, χ ≠ 1 → (ψ χ).IsNonprincipal := by | |
| let : CommGroup (L ≃ₐ[K] L) := IsCyclic.commGroup | |
| let : IsAbelianGalois K L := { is_comm.comm := mul_comm } | |
| obtain ⟨m, N, hN, hdN, hm, hnorm, hlocal, hur⟩ := | |
| exists_integer_even_artin_modulus (K := K) (L := L) | |
| obtain ⟨ψ, hid, hψ⟩ := exists_cyclic_ray_family K L N hN | |
| (idealCharacter_kernelLaw_of_even_modulus m hm hnorm hur N hlocal) | |
| exact ⟨N, hN, hdN, ψ, hid, hψ⟩ | |
| end DukePrimeDegree | |
| end | |
| end OAI | |