| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ClassFieldCongruences |
| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.HeckeFactors |
| import ClassFieldTheory.AlgebraicNumberTheory.RayClass.PrimeGeneration |
| import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.MathlibFrobeniusHilbertComparison |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| Prime-to-modulus ideals are generated by primes outside the modulus. |
| At unramified primes the arithmetic Artin element agrees with the |
| Frobenius used by the ideal-character Dirichlet series. Thus principal |
| ray ideals lie in the kernel of the actual ideal character. |
| -/ |
|
|
| noncomputable section |
| open scoped NumberField Classical |
|
|
| namespace DukePrimeDegree |
|
|
| open GlobalClassFieldTheory GlobalClassFieldTheory.GlobalClassFields |
| GlobalClassFieldTheory.IdealClassFieldTheory |
| 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] |
|
|
| /-- Agreement at unramified primes identifies the genuine ideal |
| character with composition through the arithmetic Artin map. -/ |
| theorem idealCharacter_eq_artin_comp |
| (m : RayClass.Modulus K) |
| (hm : 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) |
| (χ : (L ≃ₐ[K] L) →* ℂˣ) : |
| (idealHom K L χ).comp (RayClass.primeToModulusIdeals m).subtype = |
| χ.comp (arithmeticIdealArtinGaloisMap (K := K) (L := L) m hm) := by |
| apply RayClass.primeToModulusIdeals_hom_ext m |
| intro v hv |
| let Q := primeAbove K L v |
| let w : IsDedekindDomain.HeightOneSpectrum (𝓞 L) := |
| ⟨Q.val, Q.property.1, primeAbove_ne_bot K L v⟩ |
| have hw : w.asIdeal.LiesOver v.asIdeal := Q.property.2 |
| have hwu : Algebra.IsUnramifiedAt (𝓞 K) w.asIdeal := hunram v hv |
| change idealHom K L χ (FractionalIdealGroup.prime v) = |
| χ (arithmeticIdealArtinGaloisMap (K := K) (L := L) m hm |
| (RayClass.primeToModulusIdeal m v hv)) |
| rw [arithmeticIdealArtinGaloisMap_primeIdeal_eq_arithmeticFinitePlacePrimeArtin, |
| ClassFieldTheory.GlobalClassFieldComparison.arithmeticPrimeArtin_eq_arithmeticFrobeniusAt v w hw hwu] |
| change idealHom K L χ (integralIdealFractionalUnit v.asIdeal v.ne_bot) = _ |
| rw [idealHom_prime] |
| rfl |
|
|
| /-- No extra normalization hypothesis remains for a principal ray ideal. -/ |
| theorem idealCharacter_principalRay_eq_one |
| (m : RayClass.Modulus K) |
| (hm : 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) |
| (χ : (L ≃ₐ[K] L) →* ℂˣ) |
| (I : RayClass.primeToModulusIdeals m) |
| (hI : I ∈ RayClass.principalRayIdealSubgroup m) : |
| idealHom K L χ I = 1 := by |
| have h := DFunLike.congr_fun (idealCharacter_eq_artin_comp m hm hunram χ) I |
| change idealHom K L χ I = χ (arithmeticIdealArtinGaloisMap m hm I) at h |
| rw [arithmetic_artin_eq_one_of_principal_ray m hm I hI, map_one] at h |
| exact h |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
