import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ClassFieldCongruences
import OAI.NumberTheory.DukePrimeDegree.Arithmetic.HeckeFactors
import ClassFieldTheory.AlgebraicNumberTheory.RayClass.PrimeGeneration
import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.MathlibFrobeniusHilbertComparison

namespace OAI

/-!
# The arithmetic Artin map realizes the actual ideal character

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