| import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.ArithmeticIdealArtin |
| import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.AbelianConductorExactness |
|
|
| namespace OAI |
|
|
| /-! |
| # The even Artin kernel for totally real extensions |
|
|
| The ordinary finite congruence modulus needed by even Hecke characters is |
| obtained from the actual class-field conductor, whose infinite part is |
| empty in a totally real extension. |
| -/ |
|
|
| open scoped NumberField |
|
|
| noncomputable section |
|
|
| namespace DukePrimeDegree |
|
|
| open GlobalClassFieldTheory GlobalClassFieldTheory.GlobalClassFields |
| GlobalClassFieldTheory.IdealClassFieldTheory |
|
|
| /-- The norm subgroup of a totally real abelian extension has a defining |
| modulus with no infinite places. -/ |
| theorem exists_even_artin_modulus |
| (K L : Type) [Field K] [NumberField K] [Field L] [NumberField L] |
| [Algebra K L] [IsAbelianGalois K L] [NumberField.IsTotallyReal L] : |
| ∃ m : RayClass.Modulus K, m.infinitePart = ∅ ∧ |
| m.congruenceSubgroup ≤ (ideleClassNorm K L).range := by |
| classical |
| let H := ideleClassNormConductorialSubgroup (K := K) (L := L) |
| refine ⟨H.fullConductor, ?_, H.fullConductor_isDefiningModulus⟩ |
| change (ideleClassNormConductorialSubgroup (K := K) (L := L)).fullConductor.infinitePart = ∅ |
| rw [ideleClassNormFullConductor_infinitePart_eq_realRamificationLocus] |
| apply Finset.filter_eq_empty_iff.mpr |
| intro v _ hv |
| apply hv |
| intro w _ |
| exact (NumberField.IsTotallyReal.isReal w).isUnramified K |
|
|
| /-- Arithmetic normalization of the Artin map still annihilates every |
| principal ray ideal. -/ |
| theorem arithmetic_artin_eq_one_of_principal_ray |
| {K L : Type} [Field K] [NumberField K] [Field L] [NumberField L] |
| [Algebra K L] [IsAbelianGalois K L] |
| (m : RayClass.Modulus K) |
| (hm : m.congruenceSubgroup ≤ (ideleClassNorm K L).range) |
| (I : RayClass.primeToModulusIdeals m) |
| (hI : I ∈ RayClass.principalRayIdealSubgroup m) : |
| arithmeticIdealArtinGaloisMap (K := K) (L := L) m hm I = 1 := by |
| apply MonoidHom.mem_ker.mp |
| rw [arithmeticIdealArtinGaloisMap_ker] |
| exact principalRayIdealSubgroup_le_idealArtinKernel m |
| (ideleClassNorm K L).range hm hI |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
