| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.ZetaOrder |
| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.PermutationEuler |
|
|
| namespace OAI |
|
|
| /-! |
| # Conjugate subfields have the same zeta zero order |
|
|
| Conjugate subgroups have the same permutation character. The |
| character-to-Euler identity identifies their actual incomplete Dedekind |
| zeta functions, and analytic continuation identifies their zero orders. |
| -/ |
|
|
| noncomputable section |
| open Module Filter Set |
| open scoped NumberField BigOperators Classical Topology |
|
|
| namespace DukePrimeDegree |
|
|
| open PrimeDegreePackets PrimeDegreePackets.GaloisIdealCharacter |
| PrimeDegreePackets.HeckeMellin |
|
|
| theorem permutation_character_conjugate_subgroup |
| {G : Type*} [Group G] [Fintype G] (H : Subgroup G) (a g : G) : |
| (Representation.ofMulAction ℂ G (G ⧸ H.map (MulAut.conj a).toMonoidHom)).character g = |
| (Representation.ofMulAction ℂ G (G ⧸ H)).character g := by |
| let H' := H.map (MulAut.conj a).toMonoidHom |
| have hcard : Fintype.card H' = Fintype.card H := by |
| exact Fintype.card_congr (H.equivMapOfInjective |
| (MulAut.conj a).toMonoidHom (MulAut.conj a).injective).toEquiv.symm |
| have hsum : (∑ x : G, if x⁻¹ * g * x ∈ H' then (1 : ℂ) else 0) = |
| ∑ x : G, if x⁻¹ * g * x ∈ H then (1 : ℂ) else 0 := by |
| apply Fintype.sum_equiv (Equiv.mulRight a) |
| intro x |
| have hmem : x⁻¹ * g * x ∈ H' ↔ (x * a)⁻¹ * g * (x * a) ∈ H := by |
| rw [Subgroup.mem_map_equiv] |
| simp only [MulAut.conj_symm_apply, mul_inv_rev, mul_assoc] |
| change (if x⁻¹ * g * x ∈ H' then (1 : ℂ) else 0) = |
| if (x * a)⁻¹ * g * (x * a) ∈ H then 1 else 0 |
| rw [hmem] |
| rw [conjugacy_indicator_sum_eq_card_mul_character, |
| conjugacy_indicator_sum_eq_card_mul_character, hcard] at hsum |
| exact mul_left_cancel₀ (by exact_mod_cast Fintype.card_ne_zero (α := H)) hsum |
| |
| theorem permutation_euler_eq_of_character_eq |
| {G : Type*} [Group G] [Fintype G] (H J : Subgroup G) |
| (hχ : ∀ g, (Representation.ofMulAction ℂ G (G ⧸ H)).character g = |
| (Representation.ofMulAction ℂ G (G ⧸ J)).character g) |
| (g : G) (z : ℂ) : |
| (LinearMap.id - z • Representation.ofMulAction ℂ G (G ⧸ H) g).det = |
| (LinearMap.id - z • Representation.ofMulAction ℂ G (G ⧸ J) g).det := by |
| let : Fintype (Subgroup G) := Fintype.ofFinite _ |
| let p : Subgroup G → ℕ := fun S => if S = H then 1 else 0 |
| let q : Subgroup G → ℕ := fun S => if S = J then 1 else 0 |
| have h := euler_det_products_eq_of_characters |
| (V := fun S : Subgroup G => MonoidAlgebra ℂ (G ⧸ S)) |
| (fun S => Representation.ofMulAction ℂ G (G ⧸ S)) p q |
| (fun g => by simpa [p, q] using hχ g) g z |
| simpa [p, q] using h |
| |
| variable {F E : Type*} [Field F] [NumberField F] [Field E] [NumberField E] |
| variable [Algebra F E] [FiniteDimensional F E] [IsGalois F E] |
| local instance : Fintype (E ≃ₐ[F] E) := Fintype.ofFinite _ |
| |
| theorem cosetRationalEuler_eq_of_permutation_characters |
| (L M : IntermediateField F E) |
| (hχ : ∀ g, |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ L.fixingSubgroup)).character g = |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ M.fixingSubgroup)).character g) |
| (N : ℕ) (p : Nat.Primes) (s : ℂ) : |
| cosetRationalEuler F E L N p s = cosetRationalEuler F E M N p s := by |
| by_cases hp : p.val ∣ N |
| · simp only [cosetRationalEuler, ite_eq_left hp] |
| simp only [cosetRationalEuler, ite_eq_right hp] |
| apply Finset.prod_congr rfl |
| intro P _ |
| congr 1 |
| rw [← permutation_euler_det_eq, ← permutation_euler_det_eq] |
| exact permutation_euler_eq_of_character_eq L.fixingSubgroup M.fixingSubgroup hχ _ _ |
| |
| theorem coset_real_eq_of_permutation_characters |
| (L M : IntermediateField F E) |
| [NumberField.IsTotallyReal L] [NumberField.IsTotallyReal M] |
| (hχ : ∀ g, |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ L.fixingSubgroup)).character g = |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ M.fixingSubgroup)).character g) |
| {N : ℕ} (hN : 0 < N) (hD : (NumberField.discr E).natAbs ∣ N) |
| (s : ℝ) (hs : 1 < s) : |
| cosetMeromorphic (canonicalRealEmbeddings L) (degreePred_spec L) N s = |
| cosetMeromorphic (canonicalRealEmbeddings M) (degreePred_spec M) N s := by |
| have hs' : 1 < (s : ℂ).re := hs |
| have hL := (actual_coset_meromorphic_pairing F E L (canonicalRealEmbeddings L) |
| (degreePred_spec L) N hN hD).2.1 (s : ℂ) hs' |
| have hM := (actual_coset_meromorphic_pairing F E M (canonicalRealEmbeddings M) |
| (degreePred_spec M) N hN hD).2.1 (s : ℂ) hs' |
| exact hL.unique (hM.congr_fun (fun p => |
| cosetRationalEuler_eq_of_permutation_characters L M hχ N p s)) |
|
|
| theorem zetaZeroOrder_eq_of_permutation_characters |
| (L M : IntermediateField F E) |
| [NumberField.IsTotallyReal L] [NumberField.IsTotallyReal M] |
| (hχ : ∀ g, |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ L.fixingSubgroup)).character g = |
| (Representation.ofMulAction ℂ (E ≃ₐ[F] E) ((E ≃ₐ[F] E) ⧸ M.fixingSubgroup)).character g) |
| {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) : |
| zetaZeroOrder L β = zetaZeroOrder M β := by |
| let N := (NumberField.discr E).natAbs |
| have hN : 0 < N := Int.natAbs_pos.mpr (NumberField.discr_ne_zero E) |
| let σ := canonicalRealEmbeddings L |
| let τ := canonicalRealEmbeddings M |
| have heq := analytic_off_zero_one_eq_of_real_eq |
| (coset_analytic_off_zero_one σ (degreePred_spec L) N) |
| (coset_analytic_off_zero_one τ (degreePred_spec M) N) |
| (coset_real_eq_of_permutation_characters L M hχ hN (dvd_refl _)) |
| have hopen : IsOpen ({0, 1} : Set ℂ)ᶜ := |
| (Set.Finite.isClosed (by simp)).isOpen_compl |
| 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 hv : cosetMeromorphic σ (degreePred_spec L) N =ᶠ[𝓝[≠] (β : ℂ)] |
| cosetMeromorphic τ (degreePred_spec M) 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 |
| rw [zetaZeroOrder_eq_coset_order σ (degreePred_spec L) N hβ0 hβ1, |
| zetaZeroOrder_eq_coset_order τ (degreePred_spec M) N hβ0 hβ1] |
| exact meromorphicOrderAt_congr hv |
| |
| theorem zetaZeroOrder_fixedField_conjugate |
| [NumberField.IsTotallyReal E] |
| (H : Subgroup (E ≃ₐ[F] E)) (a : E ≃ₐ[F] E) |
| {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) : |
| let : NumberField.IsTotallyReal (IntermediateField.fixedField H) := |
| NumberField.IsTotallyReal.of_algebra _ E |
| letI : NumberField.IsTotallyReal |
| (IntermediateField.fixedField (H.map (MulAut.conj a).toMonoidHom)) := |
| NumberField.IsTotallyReal.of_algebra _ E |
| zetaZeroOrder (IntermediateField.fixedField H) β = |
| zetaZeroOrder (IntermediateField.fixedField (H.map (MulAut.conj a).toMonoidHom)) β := by |
| let : NumberField.IsTotallyReal (IntermediateField.fixedField H) := |
| NumberField.IsTotallyReal.of_algebra _ E |
| let : NumberField.IsTotallyReal |
| (IntermediateField.fixedField (H.map (MulAut.conj a).toMonoidHom)) := |
| NumberField.IsTotallyReal.of_algebra _ E |
| apply zetaZeroOrder_eq_of_permutation_characters _ _ _ hβ0 hβ1 |
| intro g |
| rw [IntermediateField.fixingSubgroup_fixedField, |
| IntermediateField.fixingSubgroup_fixedField] |
| exact (permutation_character_conjugate_subgroup H a g).symm |
| |
| end DukePrimeDegree |
| |
| end |
| |
| end OAI |
| |
