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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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import OAI.NumberTheory.DukePrimeDegree.Arithmetic.StarkDescent
import OAI.NumberTheory.DukePrimeDegree.Arithmetic.StarkScale
import OAI.NumberTheory.DukePrimeDegree.Arithmetic.StarkZeros
namespace OAI
/-!
# Assembling the normal-closure argument
Realizing cyclic Frobenius characters by even ray characters gives
zero propagation to the normal closure. Simplicity near one and
odd-index descent then exclude an exceptional real zero.
-/
noncomputable section
open Module Set IntermediateField
open scoped NumberField Classical
namespace DukePrimeDegree
open PrimeDegreePackets PrimeDegreePackets.HeckeMellin PrimeDegreePackets.AnalyticGrowth
universe u
lemma fixedFieldMultiplicity_fixingSubgroup
{K E : Type*} [Field K] [NumberField K] [NumberField.IsTotallyReal K]
[Field E] [NumberField E] [NumberField.IsTotallyReal E] [IsGalois ℚ E]
(f : K →ₐ[ℚ] E) {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) :
fixedFieldMultiplicity β f.fieldRange.fixingSubgroup = zetaMultiplicity K β := by
let L := f.fieldRange
let e : IntermediateField.fixedField L.fixingSubgroup ≃ₐ[ℚ] L :=
IntermediateField.equivOfEq (IsGalois.fixedField_fixingSubgroup L)
exact (zetaMultiplicity_algEquiv e hβ0 hβ1).trans
(zetaMultiplicity_algEquiv f.equivFieldRange hβ0 hβ1).symm
lemma fixingSubgroup_index_of_embedding
{K E : Type*} [Field K] [NumberField K] [Field E] [NumberField E]
[IsGalois ℚ E] (f : K →ₐ[ℚ] E) :
f.fieldRange.fixingSubgroup.index = finrank ℚ K := by
rw [← IntermediateField.finrank_eq_fixingSubgroup_index]
exact f.equivFieldRange.toLinearEquiv.finrank_eq.symm
/-- The zero-exclusion consequence of cyclic reciprocity, for the exact
family of normal closures required by the application. -/
theorem exceptionalRealZeroExclusion_of_ray_realizations
(d : ℕ) (hodd : Odd (d + 1))
(hray : ∀ (E : Type u) [Field E] [NumberField E] [IsGalois ℚ E]
[NumberField.IsTotallyReal E], HasCyclicRayRealizations E) :
ExceptionalRealZeroExclusion.{u} d := by
obtain ⟨a, T, ha, ha1, hT, hsimple⟩ := normalClosure_simple_zero_scaled d
obtain ⟨b, hb, hQ⟩ := rational_zetaMultiplicity_zero_scaled
refine ⟨min a b, T, lt_min ha hb, hT, ?_⟩
intro K _ _ _ hn σ hL β hβ hβ1 hzero
have hLp : 0 < Real.log |(NumberField.discr K : ℝ)| := lt_of_lt_of_le (by linarith) hL
have hfrac : min a b / Real.log |(NumberField.discr K : ℝ)| ≤ 1 / 4 := by
have ha' : min a b ≤ 1 := (min_le_left _ _).trans ha1
apply (div_le_iff₀ hLp).mpr
have hL4 := hT.trans hL
nlinarith
have hβ0 : 0 < β := by linarith
let E := normalClosure ℚ K (AlgebraicClosure K)
let : Algebra.IsAlgebraic ℚ (AlgebraicClosure K) := Algebra.IsAlgebraic.trans ℚ K _
let : IsAlgClosure ℚ (AlgebraicClosure K) := ⟨inferInstance, inferInstance⟩
let : Nonempty (K →ₐ[ℚ] AlgebraicClosure K) := ⟨IsScalarTower.toAlgHom ℚ K _⟩
let : IsNormalClosure ℚ K E := isNormalClosure_normalClosure ℚ K (AlgebraicClosure K)
let : Normal ℚ E := IsNormalClosure.normal (K := K)
let : IsGalois ℚ E := {}
let : NumberField E := {}
let : NumberField.IsTotallyReal E := normalClosure_isTotallyReal
obtain ⟨f⟩ := IntermediateField.nonempty_algHom_of_splits
(fun x : K => ⟨Algebra.IsIntegral.isIntegral x,
IsNormalClosure.splits (F := ℚ) (K := K) (L := E) x⟩)
let H := f.fieldRange.fixingSubgroup
have hHodd : Odd H.index := by
rw [fixingSubgroup_index_of_embedding f, hn]
exact hodd
have hKpos : 0 < zetaMultiplicity K β := by
rw [zetaMultiplicity_eq_zeroMultiplicity σ hn hβ0 hβ1]
exact lt_of_le_of_ne (zeroMultiplicity_nonneg (completedThetaXi_entire σ hn) _)
(Ne.symm ((zeroMultiplicity_ne_zero_iff (completedThetaXi_entire σ hn)
(dedekindXiZeros_ne_zero_totallyReal σ hn) (β : ℂ)).mpr hzero))
have hHpos : 0 < fixedFieldMultiplicity β H := by
rw [fixedFieldMultiplicity_fixingSubgroup f hβ0 hβ1]
exact hKpos
have hEpos := extension_multiplicity_pos_of_ray_realizations (hray E) hβ0 hβ1 H hHpos
have hβa : 1 - a / Real.log |(NumberField.discr K : ℝ)| ≤ β := by
have hh := div_le_div_of_nonneg_right (min_le_left a b) hLp.le
linarith
have hEone := hsimple σ hn hL β hβ0 hβ1 hβa (ne_of_gt hEpos)
have hβb : 1 - b / Real.log |(NumberField.discr K : ℝ)| ≤ β := by
have hh := div_le_div_of_nonneg_right (min_le_right a b) hLp.le
linarith
exact no_odd_index_zero_of_ray_realizations (hray E) hβ0 hβ1 hEone H hHodd hHpos
(hQ _ (hT.trans hL) β ⟨hβb, hβ1.le⟩ hβ0 hβ1)
end DukePrimeDegree
end
end OAI
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