| 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 | |