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b3ddf75 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
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