File size: 3,406 Bytes
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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ClassFieldBridge
namespace OAI
/-!
# A common even modulus away from ramification
Enlarging the finite conductor to contain the discriminant primes leaves
its infinite part empty and its Artin-kernel property intact. An ordinary
positive integer then imposes all of its local congruences.
-/
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] [IsAbelianGalois K L]
variable [NumberField.IsTotallyReal L]
theorem exists_even_unramified_artin_modulus :
∃ m : RayClass.Modulus K, m.infinitePart = ∅ ∧
m.congruenceSubgroup ≤ (ideleClassNorm K L).range ∧
∀ v : IsDedekindDomain.HeightOneSpectrum (𝓞 K), v ∉ m.finitePart.support →
Algebra.IsUnramifiedAt (𝓞 K) (primeAbove K L v).val := by
obtain ⟨m, hm, hnorm⟩ := exists_even_artin_modulus K L
let D : 𝓞 K := (NumberField.discr L).natAbs
have hD : D ≠ 0 := by
change ((NumberField.discr L).natAbs : 𝓞 K) ≠ 0
exact_mod_cast (Int.natAbs_pos.mpr (NumberField.discr_ne_zero L)).ne'
let I := integralIdealFractionalUnit (Ideal.span {D})
(by simpa only [ne_eq, Ideal.span_singleton_eq_bot] using hD)
let b : RayClass.FiniteModulus K :=
(FractionalIdealGroup.countVector I).mapRange Int.natAbs rfl
let m' := m.replaceFinitePart (m.finitePart + b)
have hle : m ≤ m' := by
constructor
· exact le_add_of_nonneg_right (by positivity)
· exact Finset.Subset.refl _
refine ⟨m', hm, (RayClass.Modulus.congruenceSubgroup_antitone hle).trans hnorm, ?_⟩
intro v hv
have hv0 : m'.finitePart v = 0 := Finsupp.notMem_support_iff.mp hv
have hb : b v = 0 := by
change m.finitePart v + b v = 0 at hv0
omega
have hcount : FractionalIdeal.count K v (Ideal.span {D}) = 0 := by
have hc : Int.natAbs (FractionalIdeal.count K v (Ideal.span {D})) = 0 := hb
exact Int.natAbs_eq_zero.mp hc
have hnot : D ∉ v.asIdeal := by
exact mt ((principal_count_ne_zero_iff_mem K D hD v).mpr) (by simpa using hcount)
let : (primeAbove K L v).val.IsPrime := (primeAbove K L v).property.1
exact unramified_of_absDiscr_not_mem K L v.asIdeal (primeAbove K L v).val
(primeAbove K L v).property.2.over.symm hnot
theorem exists_integer_even_artin_modulus :
∃ (m : RayClass.Modulus K) (N : ℕ), 0 < N ∧
(NumberField.discr L).natAbs ∣ N ∧ m.infinitePart = ∅ ∧
m.congruenceSubgroup ≤ (ideleClassNorm K L).range ∧
(∀ v ∈ m.finitePart.support,
algebraMap (𝓞 K) (v.adicCompletionIntegers K) (N : 𝓞 K) ∈
IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ m.finitePart v) ∧
∀ v : IsDedekindDomain.HeightOneSpectrum (𝓞 K), v ∉ m.finitePart.support →
Algebra.IsUnramifiedAt (𝓞 K) (primeAbove K L v).val := by
obtain ⟨m, hm, hnorm, hur⟩ := exists_even_unramified_artin_modulus (K := K) (L := L)
obtain ⟨N, hN, hdN, hlocal⟩ := exists_integer_for_local_congruences
m.finitePart.support m.finitePart (Int.natAbs_pos.mpr (NumberField.discr_ne_zero L))
exact ⟨m, N, hN, hdN, hm, hnorm, hlocal, hur⟩
end DukePrimeDegree
end
end OAI
|