| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.CyclicRealCharacters |
| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.PermutationEuler |
|
|
| namespace OAI |
|
|
| /-! |
| # Cyclic subfield factors |
|
|
| For an intermediate field in a cyclic extension, the incomplete zeta |
| function is the product of exactly those Hecke factors whose characters |
| are trivial on the fixing subgroup. These subfield orders determine the |
| real character selected by a simple zero, without general Artin induction. |
| -/ |
|
|
| noncomputable section |
| open Module Filter Set |
| open scoped NumberField BigOperators Classical Topology |
|
|
| namespace DukePrimeDegree |
|
|
| open PrimeDegreePackets PrimeDegreePackets.GaloisIdealCharacter |
| PrimeDegreePackets.HeckeMellin |
|
|
| section Quotient |
|
|
| variable {C : Type*} [CommGroup C] [Finite C] |
| local instance (D : Subgroup C) : Fintype (C ⧸ D →* ℂˣ) := Fintype.ofFinite _ |
|
|
| theorem quotient_translation_euler (D : Subgroup C) (g : C) (z : ℂ) : |
| (LinearMap.id - z • |
| (LinearEquiv.funCongrLeft ℂ ℂ (MulAction.toPerm (β := C ⧸ D) g⁻¹)).toLinearMap).det = |
| ∏ η : C ⧸ D →* ℂˣ, (1 - z * (η (QuotientGroup.mk g) : ℂ)) := by |
| let b := (complexMulBasis (C ⧸ D)).reindex (Equiv.inv (C ⧸ D →* ℂˣ)) |
| have hb (η : C ⧸ D →* ℂˣ) (q : C ⧸ D) : b η q = ((η⁻¹) q : ℂ) := by |
| simp [b, complexMulBasis_apply] |
| apply euler_det_eq_prod_of_eigenbasis b |
| intro η |
| funext q |
| change b η (g⁻¹ • q) = (η (QuotientGroup.mk g) : ℂ) * b η q |
| rw [hb, hb] |
| change ((η⁻¹) ((QuotientGroup.mk g)⁻¹ * q) : ℂ) = _ |
| simp only [map_mul, map_inv, MonoidHom.inv_apply, inv_inv, Units.val_mul] |
|
|
| theorem sum_quotient_orders_of_unique_character |
| (D : Subgroup C) (m : (C →* ℂˣ) → WithTop ℤ) (χ : C →* ℂˣ) |
| (hχ : m χ = 1) (hrest : ∀ φ, φ ≠ χ → m φ = 0) : |
| (∑ η : C ⧸ D →* ℂˣ, m (η.comp (QuotientGroup.mk' D))) = |
| if D ≤ χ.ker then 1 else 0 := by |
| by_cases hD : D ≤ χ.ker |
| · rw [ite_eq_left hD] |
| let η₀ := QuotientGroup.lift D χ hD |
| have he : η₀.comp (QuotientGroup.mk' D) = χ := rfl |
| rw [Finset.sum_eq_single η₀] |
| · rw [he, hχ] |
| · intro η _ hη |
| apply hrest |
| intro h |
| apply hη |
| apply MonoidHom.ext |
| intro q |
| induction q using Quotient.inductionOn |
| rename_i c |
| exact congrArg (fun φ : C →* ℂˣ => φ c) h |
| · simp |
| · rw [ite_eq_right hD] |
| apply Finset.sum_eq_zero |
| intro η _ |
| apply hrest |
| intro h |
| apply hD |
| intro c hc |
| change χ c = 1 |
| rw [← h] |
| change η (QuotientGroup.mk c) = 1 |
| rw [(QuotientGroup.eq_one_iff c).mpr hc, map_one] |
|
|
| end Quotient |
|
|
| variable {F E : Type*} [Field F] [NumberField F] [Field E] [NumberField E] |
| variable [Algebra F E] [FiniteDimensional F E] [IsGalois F E] |
| variable [IsCyclic (E ≃ₐ[F] E)] |
| attribute [local instance] IsCyclic.commGroup |
| local instance (L : IntermediateField F E) : |
| Fintype ((E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ) := Fintype.ofFinite _ |
|
|
| theorem cyclic_subfield_local_euler |
| (L : IntermediateField F E) {N : ℕ} |
| (p : Nat.Primes) {s : ℂ} (hs : 1 < s.re) : |
| cosetRationalEuler F E L N p s = |
| ∏ η : (E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ, |
| ∏ P : idealsOverPrime F p.val, |
| (1 - ArtinSeries.idealTerm |
| (η.comp (QuotientGroup.mk' L.fixingSubgroup)) N hs P.val)⁻¹ := by |
| by_cases hp : p.val ∣ N |
| · simp only [cosetRationalEuler, ite_eq_left hp, |
| ArtinSeries.local_euler_bad F E _ p hp hs, Finset.prod_const_one] |
| rw [cosetRationalEuler, ite_eq_right hp, Finset.prod_comm] |
| apply Finset.prod_congr rfl |
| intro P _ |
| let PP := rationalPrimeSpectrum F p P |
| have hP := (mem_idealsOverPrime_iff F p.property.ne_zero P.val).mp P.property |
| have hc : IsCoprime P.val (ThetaConstruction.rationalModulusIdeal (𝓞 F) N) := by |
| apply Ideal.isCoprime_iff_codisjoint.mpr |
| apply Codisjoint.mono ((Ideal.span_singleton_le_iff_mem _).mpr hP.2) le_rfl |
| exact ((Ideal.isCoprime_span_singleton_iff _ _).mpr |
| (p.property.coprime_iff_not_dvd.mpr hp).cast).codisjoint |
| have he (η : (E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ) : |
| ArtinSeries.idealTerm (η.comp (QuotientGroup.mk' L.fixingSubgroup)) N hs P.val = |
| (η (QuotientGroup.mk (primeFrobenius F E PP)) : ℂ) * |
| (Ideal.absNorm P.val : ℂ)^(-s) := by |
| rw [ArtinSeries.idealTerm_apply] |
| change artinCoefficient F E (η.comp (QuotientGroup.mk' L.fixingSubgroup)) N |
| PP.asIdeal * _ = _ |
| have hc' : IsCoprime PP.asIdeal (ThetaConstruction.rationalModulusIdeal (𝓞 F) N) := hc |
| rw [artinCoefficient, dite_eq_left PP.ne_bot, ite_eq_left hc', idealHom_prime] |
| rfl |
| rw [quotient_translation_euler, ← Finset.prod_inv_distrib] |
| apply Finset.prod_congr rfl |
| intro η _ |
| rw [he] |
| congr 2 |
| ring |
|
|
| theorem cyclic_subfield_artin_product |
| (L : IntermediateField F E) {N : ℕ} (hN : 0 < N) |
| (hD : (NumberField.discr E).natAbs ∣ N) {s : ℂ} (hs : 1 < s.re) : |
| NumberField.dedekindZeta L s * removedEulerPolynomial L N s = |
| ∏ η : (E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ, |
| artinDirichlet F E (η.comp (QuotientGroup.mk' L.fixingSubgroup)) N s := by |
| have hA := hasProd_prod |
| (s := (Finset.univ : Finset ((E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ))) |
| (fun η _ => ArtinSeries.dirichlet_hasProd (F := F) (E := E) |
| (η.comp (QuotientGroup.mk' L.fixingSubgroup)) N hs) |
| exact (cosetRationalEuler_hasProd F E L N hN hD hs).unique |
| (hA.congr_fun (fun p => cyclic_subfield_local_euler L p hs)) |
|
|
| variable [NumberField.IsTotallyReal F] [NumberField.IsTotallyReal E] |
| variable {d e N : ℕ} [NeZero N] |
|
|
| omit [NumberField.IsTotallyReal E] in |
| theorem cyclic_subfield_sum_orders |
| (L : IntermediateField F E) [NumberField.IsTotallyReal L] |
| (σ : OrderedEmbeddings (d + 1) F) (hn : finrank ℚ F = d + 1) |
| (τ : OrderedEmbeddings (e + 1) L) (he : finrank ℚ L = e + 1) |
| (ψ : ((E ≃ₐ[F] E) →* ℂˣ) → EvenRayCharacter F N) |
| (hid : ∀ χ, (ψ χ).idealHom = idealHom F E χ) |
| (hψ : ∀ χ, χ ≠ 1 → (ψ χ).IsNonprincipal) |
| (hD : (NumberField.discr E).natAbs ∣ N) |
| {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) : |
| (∑ η : (E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ, |
| meromorphicOrderAt (cyclicHeckeFactor σ hn ψ |
| (η.comp (QuotientGroup.mk' L.fixingSubgroup))) (β : ℂ)) = |
| meromorphicOrderAt (completedThetaXi τ he) (β : ℂ) := by |
| let χ := fun η : (E ≃ₐ[F] E) ⧸ L.fixingSubgroup →* ℂˣ => |
| η.comp (QuotientGroup.mk' L.fixingSubgroup) |
| have hp : AnalyticOnNhd ℂ (fun z => ∏ η, cyclicHeckeFactor σ hn ψ (χ η) z) |
| ({0, 1} : Set ℂ)ᶜ := by |
| intro z hz |
| exact Finset.analyticAt_fun_prod Finset.univ |
| (fun η _ => cyclicHeckeFactor_analytic σ hn ψ (χ η) z hz) |
| have hq : AnalyticOnNhd ℂ (cosetMeromorphic τ he N) ({0, 1} : Set ℂ)ᶜ := by |
| intro z hz |
| rw [cosetMeromorphic_eq_factor] |
| exact (cosetXiFactor_analyticAt N (by simpa using (not_or.mp hz).1) |
| (by simpa using (not_or.mp hz).2)).mul |
| ((completedThetaXi_entire τ he).analyticAt z) |
| have heq := analytic_off_zero_one_eq_of_real_eq hp hq (fun s hs => by |
| rw [cosetMeromorphic_eq_dedekindZeta τ he N hs, |
| cyclic_subfield_artin_product L (NeZero.pos N) hD hs] |
| exact Finset.prod_congr rfl (fun η _ => |
| cyclicHeckeFactor_eq_artinDirichlet σ hn ψ hid hψ (χ η) hs)) |
| have hβ : (β : ℂ) ∈ ({0, 1} : Set ℂ)ᶜ := by |
| simp only [mem_compl_iff, mem_insert_iff, mem_singleton_iff, not_or] |
| exact ⟨Complex.ofReal_ne_zero.mpr hβ0.ne', by exact_mod_cast hβ1.ne⟩ |
| have hopen : IsOpen ({0, 1} : Set ℂ)ᶜ := |
| (Set.Finite.isClosed (by simp)).isOpen_compl |
| have hv : (fun z => ∏ η, cyclicHeckeFactor σ hn ψ (χ η) z) =ᶠ[𝓝[≠] (β : ℂ)] |
| cosetMeromorphic τ he N := by |
| have hu : ∀ᶠ z in 𝓝 (β : ℂ), z ∈ ({0, 1} : Set ℂ)ᶜ := hopen.mem_nhds hβ |
| filter_upwards [hu.filter_mono nhdsWithin_le_nhds] with z hz |
| exact heq hz |
| have H := meromorphicOrderAt_congr hv |
| rw [meromorphicOrderAt_fun_prod (s := Finset.univ) |
| (f := fun η => cyclicHeckeFactor σ hn ψ (χ η)) |
| (fun η _ => (cyclicHeckeFactor_analytic σ hn ψ (χ η) (β : ℂ) hβ).meromorphicAt), |
| cosetMeromorphic_order_eq τ he N hβ0 hβ1] at H |
| exact H |
|
|
| omit [NumberField.IsTotallyReal E] in |
| theorem cyclic_subfield_order_of_unique_character |
| (L : IntermediateField F E) [NumberField.IsTotallyReal L] |
| (σ : OrderedEmbeddings (d + 1) F) (hn : finrank ℚ F = d + 1) |
| (τ : OrderedEmbeddings (e + 1) L) (he : finrank ℚ L = e + 1) |
| (ψ : ((E ≃ₐ[F] E) →* ℂˣ) → EvenRayCharacter F N) |
| (hid : ∀ χ, (ψ χ).idealHom = idealHom F E χ) |
| (hψ : ∀ χ, χ ≠ 1 → (ψ χ).IsNonprincipal) |
| (hD : (NumberField.discr E).natAbs ∣ N) |
| {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) |
| (χ : (E ≃ₐ[F] E) →* ℂˣ) |
| (hχ : meromorphicOrderAt (cyclicHeckeFactor σ hn ψ χ) (β : ℂ) = 1) |
| (hrest : ∀ φ, φ ≠ χ → |
| meromorphicOrderAt (cyclicHeckeFactor σ hn ψ φ) (β : ℂ) = 0) : |
| meromorphicOrderAt (completedThetaXi τ he) (β : ℂ) = |
| if L.fixingSubgroup ≤ χ.ker then 1 else 0 := by |
| rw [← cyclic_subfield_sum_orders L σ hn τ he ψ hid hψ hD hβ0 hβ1] |
| exact sum_quotient_orders_of_unique_character L.fixingSubgroup _ χ hχ hrest |
|
|
| end DukePrimeDegree |
|
|
| end |
|
|
| end OAI |
|
|
