import Mathlib import OAI.NumberTheory.QuadraticForms.CyclotomicComponents namespace OAI /-! Component length forms at finite and infinite places. -/ namespace MultiplicityCounterexample.SourceComponentLengthAtPlaces noncomputable section open TensorProduct Polynomial NumberField open LinearMap (BilinForm) open CyclicComponent CyclotomicQuotient open FixedFieldLocalComparison CyclotomicPlaceComparison CyclotomicLengthLocalComparison open TensorAlgebraModule variable (d : β„•) [NeZero d] [Fact (2 < d)] local notation "𝔽" => SourceCyclotomicField.F d local notation "𝔼" => CyclotomicQuotient.E d local notation "z" => AdjoinRoot.root (cyclotomic d β„š) def conjugation : 𝔼 ≃ₐ[β„š] 𝔼 := (IsCMField.complexConj 𝔼).restrictScalars β„š local notation "Οƒ" => conjugation d variable {V W : Type*} [AddCommGroup V] [Module β„š V] [AddCommGroup W] [Module β„š W] lemma component_adjoint (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B (T x) (T y) = B x y) : letI := componentScalarModule T p (cyclotomic d β„š) hf TraceHermitian.AdjointRule Οƒ (B.restrict (component T p)) := by let := componentScalarModule T p (cyclotomic d β„š) hf apply TraceHermitian.adjointRule_of_generator Οƒ _ z (AdjoinRoot.adjoinRoot_eq_top (f := cyclotomic d β„š)) Β· change z * IsCMField.complexConj 𝔼 z = 1 rw [SourceCyclotomicField.conj_root d] exact mul_inv_cancelβ‚€ ((root_primitive d).ne_zero (NeZero.ne d)) Β· intro x y change B ((z β€’ x : component T p) : V) ((z β€’ y : component T p) : V) = B x y have hroot (w : component T p) : ((z β€’ w : component T p) : V) = T w := congrArg Subtype.val (root_smul (componentOperator T p) (cyclotomic d β„š) (component_annihilated T p (cyclotomic d β„š) hf) w) rw [hroot, hroot] exact hT x y lemma component_selfAdjoint (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B (T x) (T y) = B x y) : letI := componentScalarModule T p (cyclotomic d β„š) hf SelfAdjointScalars (F := 𝔽) (B.restrict (component T p)) := by let := componentScalarModule T p (cyclotomic d β„š) hf exact selfAdjointScalars_of_fixed_involution Οƒ (SourceCyclotomicField.conj_fixed d) _ (component_adjoint d T B p hf hT) def componentLength (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B (T x) (T y) = B x y) : letI := componentScalarModule T p (cyclotomic d β„š) hf QuadraticForm 𝔽 (component T p) := by letI := componentScalarModule T p (cyclotomic d β„š) hf exact sourceLength (B.restrict (component T p)) (component_selfAdjoint d T B p hf hT) lemma componentLength_value (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B (T x) (T y) = B x y) (hs : B.IsSymm) (x : component T p) : letI := componentScalarModule T p (cyclotomic d β„š) hf algebraMap 𝔽 𝔼 (componentLength d T B p hf hT x) = TraceHermitian.lift (E := 𝔼) (B.restrict (component T p)) x x := by let := componentScalarModule T p (cyclotomic d β„š) hf apply sourceLength_value Οƒ (fun a => IsCMField.complexConj_apply_apply 𝔼 a) (SourceCyclotomicField.exists_scalar_of_conj_fixed d) (SourceCyclotomicField.F_finrank d) (B.restrict (component T p)) (component_selfAdjoint d T B p hf hT) ⟨fun a b => hs.eq a b⟩ (component_adjoint d T B p hf hT) omit [Fact (2 < d)] in lemma inverse_root : aeval z⁻¹ (cyclotomic d β„š) = 0 := by have h := (isRoot_cyclotomic_iff_charZero (NeZero.pos d)).mpr (root_primitive d).inv rw [IsRoot, ← map_cyclotomic d (algebraMap β„š 𝔼), ← evalβ‚‚_eq_eval_map] at h exact h lemma componentLength_nondegenerate (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B (T x) (T y) = B x y) (hs : B.IsSymm) (hn : B.Nondegenerate) (hc : IsCoprime (cyclotomic d β„š) p) : letI := componentScalarModule T p (cyclotomic d β„š) hf (componentLength d T B p hf hT).Nondegenerate := by let := componentScalarModule T p (cyclotomic d β„š) hf apply sourceLength_nondegenerate (B.restrict (component T p)) (component_selfAdjoint d T B p hf hT) ⟨fun x y => hs.eq x y⟩ apply ComponentNondegenerate.restriction_nondegenerate T B (component T p) z ((root_primitive d).ne_zero (NeZero.ne d)) _ hT hs hn p (cyclotomic d β„š) hc rfl (inverse_root d) intro x exact congrArg Subtype.val (root_smul (componentOperator T p) (cyclotomic d β„š) (component_annihilated T p (cyclotomic d β„š) hf) x) def RationalPlaceDatum (A : Type*) [CommRing A] [Algebra β„š A] (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) : Prop := βˆƒ e : (A βŠ—[β„š] V) ≃ₗ[A] (A βŠ—[β„š] W), (βˆ€ x, e (T.baseChange A x) = U.baseChange A (e x)) ∧ (βˆ€ x y, Bβ‚‚.baseChange A (e x) (e y) = B₁.baseChange A x y) omit [Fact (2 < d)] in lemma componentDatum {A : Type*} [CommRing A] [Algebra β„š A] (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (hfT : aeval T (cyclotomic d β„š * p) = 0) (hfU : aeval U (cyclotomic d β„š * p) = 0) (he : RationalPlaceDatum A T U B₁ Bβ‚‚) : letI := componentScalarModule T p (cyclotomic d β„š) hfT letI := componentScalarModule U p (cyclotomic d β„š) hfU LocalEquivariantIsometry A (B₁.restrict (component T p)) (Bβ‚‚.restrict (component U p)) z := by let := componentScalarModule T p (cyclotomic d β„š) hfT let := componentScalarModule U p (cyclotomic d β„š) hfU obtain ⟨e, hT, hB⟩ := he exact ComponentRootLocalComparison.component_root_local_datum T U B₁ Bβ‚‚ e hT hB p (cyclotomic d β„š) hfT hfU theorem componentLength_isometric_at_all_places (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (hfT : aeval T (cyclotomic d β„š * p) = 0) (hfU : aeval U (cyclotomic d β„š * p) = 0) (hT : βˆ€ x y, B₁ (T x) (T y) = B₁ x y) (hU : βˆ€ x y, Bβ‚‚ (U x) (U y) = Bβ‚‚ x y) (hreal : RationalPlaceDatum ℝ T U B₁ Bβ‚‚) (hloc : βˆ€ (q : β„•) (_ : Fact q.Prime), RationalPlaceDatum (Padic q) T U B₁ Bβ‚‚) : letI := componentScalarModule T p (cyclotomic d β„š) hfT letI := componentScalarModule U p (cyclotomic d β„š) hfU (βˆ€ v : IsDedekindDomain.HeightOneSpectrum (π“ž 𝔽), Nonempty (((componentLength d T B₁ p hfT hT).baseChange (v.adicCompletion 𝔽)).IsometryEquiv ((componentLength d U Bβ‚‚ p hfU hU).baseChange (v.adicCompletion 𝔽)))) ∧ (βˆ€ v : InfinitePlace 𝔽, Nonempty (((componentLength d T B₁ p hfT hT).baseChange v.Completion).IsometryEquiv ((componentLength d U Bβ‚‚ p hfU hU).baseChange v.Completion))) := by let := componentScalarModule T p (cyclotomic d β„š) hfT let := componentScalarModule U p (cyclotomic d β„š) hfU constructor Β· intro v exact length_isometric_at_every_finitePlace _ _ _ _ z (AdjoinRoot.adjoinRoot_eq_top (f := cyclotomic d β„š)) (fun q hq => componentDatum d T U B₁ Bβ‚‚ p hfT hfU (hloc q hq)) v Β· intro v exact length_isometric_at_every_infinitePlace _ _ _ _ z (AdjoinRoot.adjoinRoot_eq_top (f := cyclotomic d β„š)) (componentDatum d T U B₁ Bβ‚‚ p hfT hfU hreal) v end end MultiplicityCounterexample.SourceComponentLengthAtPlaces namespace MultiplicityCounterexample.CyclotomicComponentFactor noncomputable section open Polynomial variable {V : Type*} [AddCommGroup V] [Module β„š V] lemma cyclotomic_dvd (m d : β„•) (hd : d ∣ m) : cyclotomic d β„š ∣ (X^m-1 : β„š[X]) := by apply dvd_trans (cyclotomic.dvd_X_pow_sub_one d β„š) obtain ⟨k,rfl⟩ := hd simpa only [pow_mul,one_pow] using sub_dvd_pow_sub_pow (X^d : β„š[X]) 1 k theorem exists_complement (T : Module.End β„š V) (m d : β„•) (hm : 0 < m) (hd : d ∣ m) (hT : T^m=1) : βˆƒ p : β„š[X], cyclotomic d β„š*p=X^m-1 ∧ IsCoprime (cyclotomic d β„š) p ∧ aeval T (cyclotomic d β„š*p)=0 := by obtain ⟨p,hp⟩ := cyclotomic_dvd m d hd refine ⟨p,hp.symm,?_,?_⟩ Β· apply Polynomial.Separable.isCoprime rw [← hp] exact (X_pow_sub_one_separable_iff).mpr (by exact_mod_cast hm.ne') Β· rw [← hp,map_sub,map_pow,aeval_X,map_one,hT,sub_self] lemma commute_aeval (T : Module.End β„š V) (p q : β„š[X]) : Commute (aeval T p) (aeval T q) := (Commute.all p q).map (aeval T) theorem range_eq_ker (T : Module.End β„š V) (f p : β„š[X]) (hc : IsCoprime f p) (hT : aeval T (f*p)=0) : LinearMap.range (aeval T p)=LinearMap.ker (aeval T f) := by apply le_antisymm Β· rintro x ⟨y,rfl⟩ change (aeval T f) ((aeval T p) y)=0 rw [← Module.End.mul_apply,← map_mul,hT] rfl Β· intro x hx obtain ⟨a,b,h⟩ := hc refine ⟨(aeval T b) x,?_⟩ have h' := congrArg (fun q : β„š[X] => (aeval T q) x) h simp only [map_add,map_mul,map_one,LinearMap.add_apply,Module.End.mul_apply, Module.End.one_apply] at h' have hx' : (aeval T f) x=0 := hx rw [hx',map_zero,zero_add] at h' rw [← Module.End.mul_apply,(commute_aeval T p b).eq,Module.End.mul_apply] exact h' lemma component_annihilation (T : Module.End β„š V) (f p : β„š[X]) (hT : aeval T (f*p)=0) (x : LinearMap.range (aeval T p)) : (aeval T f) x=0 := by obtain ⟨y,hy⟩ := x.property rw [← hy,← Module.End.mul_apply,← map_mul,hT] rfl theorem semisimple (T : Module.End β„š V) (m : β„•) (hm : 0 < m) (hT : T^m=1) : T.IsSemisimple := by apply Module.End.isSemisimple_of_squarefree_aeval_eq_zero ((X_pow_sub_one_separable_iff (F := β„š)).mpr (by exact_mod_cast hm.ne')).squarefree rw [map_sub,map_pow,aeval_X,map_one,hT,sub_self] end end MultiplicityCounterexample.CyclotomicComponentFactor namespace MultiplicityCounterexample.HasseMinkowskiGap noncomputable section open NumberField variable {F V W : Type*} [Field F] [NumberField F] [AddCommGroup V] [Module F V] [AddCommGroup W] [Module F W] def AllPlacesIsometric (Q₁ : QuadraticForm F V) (Qβ‚‚ : QuadraticForm F W) : Prop := (βˆ€ v : IsDedekindDomain.HeightOneSpectrum (π“ž F), Nonempty ((Q₁.baseChange (v.adicCompletion F)).IsometryEquiv (Qβ‚‚.baseChange (v.adicCompletion F)))) ∧ (βˆ€ v : InfinitePlace F, Nonempty ((Q₁.baseChange v.Completion).IsometryEquiv (Qβ‚‚.baseChange v.Completion))) def NumberFieldQuadraticLocalGlobal : Prop := βˆ€ (F : Type) [Field F] [NumberField F] (V W : Type) [AddCommGroup V] [Module F V] [FiniteDimensional F V] [AddCommGroup W] [Module F W] [FiniteDimensional F W] (Q₁ : QuadraticForm F V) (Qβ‚‚ : QuadraticForm F W), Q₁.Nondegenerate β†’ Qβ‚‚.Nondegenerate β†’ AllPlacesIsometric Q₁ Qβ‚‚ β†’ Nonempty (Q₁.IsometryEquiv Qβ‚‚) end end MultiplicityCounterexample.HasseMinkowskiGap namespace MultiplicityCounterexample.SourceComponentLengthAtPlaces noncomputable section open TensorProduct Polynomial NumberField open LinearMap (BilinForm) open CyclicComponent CyclotomicQuotient HasseMinkowskiGap variable (d : β„•) [NeZero d] [Fact (2 < d)] local notation "𝔽" => SourceCyclotomicField.F d variable {V W : Type*} [AddCommGroup V] [Module β„š V] [FiniteDimensional β„š V] [AddCommGroup W] [Module β„š W] [FiniteDimensional β„š W] theorem finite_order_component_hasse_input (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (m : β„•) (hm : Odd m) (hd : d ∣ m) (hTm : T ^ m = 1) (hUm : U ^ m = 1) (hs₁ : B₁.IsSymm) (hsβ‚‚ : Bβ‚‚.IsSymm) (hn₁ : B₁.Nondegenerate) (hnβ‚‚ : Bβ‚‚.Nondegenerate) (hT : βˆ€ x y, B₁ (T x) (T y) = B₁ x y) (hU : βˆ€ x y, Bβ‚‚ (U x) (U y) = Bβ‚‚ x y) (hreal : RationalPlaceDatum ℝ T U B₁ Bβ‚‚) (hloc : βˆ€ (q : β„•) (_ : Fact q.Prime), RationalPlaceDatum (Padic q) T U B₁ Bβ‚‚) : βˆƒ (p : β„š[X]) (hfT : aeval T (cyclotomic d β„š * p) = 0) (hfU : aeval U (cyclotomic d β„š * p) = 0), cyclotomic d β„š * p = X ^ m - 1 ∧ IsCoprime (cyclotomic d β„š) p ∧ letI := componentScalarModule T p (cyclotomic d β„š) hfT letI := componentScalarModule U p (cyclotomic d β„š) hfU FiniteDimensional 𝔽 (component T p) ∧ FiniteDimensional 𝔽 (component U p) ∧ (componentLength d T B₁ p hfT hT).Nondegenerate ∧ (componentLength d U Bβ‚‚ p hfU hU).Nondegenerate ∧ AllPlacesIsometric (componentLength d T B₁ p hfT hT) (componentLength d U Bβ‚‚ p hfU hU) := by have hmpos : 0 < m := by obtain ⟨k, hk⟩ := hm; omega obtain ⟨p, hp, hc, hfT⟩ := CyclotomicComponentFactor.exists_complement T m d hmpos hd hTm have hfU : aeval U (cyclotomic d β„š * p) = 0 := by rw [hp, map_sub, map_pow, aeval_X, map_one, hUm, sub_self] refine ⟨p, hfT, hfU, hp, hc, ?_⟩ let := componentScalarModule T p (cyclotomic d β„š) hfT let := componentScalarModule U p (cyclotomic d β„š) hfU exact ⟨Module.Finite.of_restrictScalars_finite β„š 𝔽 (component T p), Module.Finite.of_restrictScalars_finite β„š 𝔽 (component U p), componentLength_nondegenerate d T B₁ p hfT hT hs₁ hn₁ hc, componentLength_nondegenerate d U Bβ‚‚ p hfU hU hsβ‚‚ hnβ‚‚ hc, componentLength_isometric_at_all_places d T U B₁ Bβ‚‚ p hfT hfU hT hU hreal hloc⟩ end end MultiplicityCounterexample.SourceComponentLengthAtPlaces namespace MultiplicityCounterexample.IdentityComponentAtPlaces noncomputable section open TensorProduct Polynomial NumberField open LinearMap (BilinForm) open CyclicComponent FixedFieldLocalComparison CyclotomicPlaceComparison open SourceComponentLengthAtPlaces HasseMinkowskiGap TensorAlgebraModule variable {V W : Type*} [AddCommGroup V] [Module β„š V] [AddCommGroup W] [Module β„š W] lemma identity_on_component (T : Module.End β„š V) (p : β„š[X]) (hf : aeval T ((X - 1) * p) = 0) (x : component T p) : T x = x := by have h := component_annihilated T p (X - 1) hf have he : componentOperator T p - 1 = 0 := by simpa only [map_sub, aeval_X, map_one] using h have hx := congrArg (fun f : Module.End β„š (component T p) => (f x : V)) he change T x - (x : V) = 0 at hx exact sub_eq_zero.mp hx lemma restriction_nondegenerate (T : Module.End β„š V) (B : BilinForm β„š V) (p : β„š[X]) (hf : aeval T ((X - 1) * p) = 0) (hc : IsCoprime (X - 1) p) (hT : βˆ€ x y, B (T x) (T y) = B x y) (hs : B.IsSymm) (hn : B.Nondegenerate) : (B.restrict (component T p)).Nondegenerate := by apply ComponentNondegenerate.restriction_nondegenerate T B (component T p) (1 : β„š) one_ne_zero _ hT hs hn p (X - 1) hc rfl (by simp) intro x simpa only [one_smul] using (identity_on_component T p hf x).symm lemma rat_selfAdjoint (B : BilinForm β„š V) : SelfAdjointScalars (F := β„š) B := by intro a x y simp only [map_smul, LinearMap.smul_apply] lemma fixedForm_rat (B : BilinForm β„š V) : fixedForm B (rat_selfAdjoint B) = B := by ext x y have h := TraceHermitian.trace_lift_one (E := β„š) B x y simpa only [Algebra.trace_self, LinearMap.id_apply, fixedForm_apply] using h lemma fixedForm_rat_canonical (B : BilinForm β„š V) : letI : Algebra β„š β„š := DivisionRing.toRatAlgebra fixedForm (F := β„š) B (rat_selfAdjoint B) = B := by exact fixedForm_rat B lemma componentDatum {A : Type*} [CommRing A] [Algebra β„š A] (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (he : RationalPlaceDatum A T U B₁ Bβ‚‚) : LocalEquivariantIsometry A (B₁.restrict (component T p)) (Bβ‚‚.restrict (component U p)) (1 : β„š) := by obtain ⟨e, hT, hB⟩ := he obtain ⟨c, _, hc⟩ := PolynomialComponentRestriction.component_local_datum T U B₁ Bβ‚‚ e hT hB p exact ⟨c, by intro x; simp only [map_one, Module.End.one_apply], hc⟩ lemma restricted_forms_at_every_finitePlace (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (hloc : βˆ€ (q : β„•) (_ : Fact q.Prime), RationalPlaceDatum (Padic q) T U B₁ Bβ‚‚) (v : IsDedekindDomain.HeightOneSpectrum (π“ž β„š)) : letI : Algebra β„š (v.adicCompletion β„š) := IsDedekindDomain.HeightOneSpectrum.instAlgebraAdicCompletion (π“ž β„š) β„š v Nonempty (((B₁.restrict (component T p)).baseChange (v.adicCompletion β„š)).IsometryEquiv ((Bβ‚‚.restrict (component U p)).baseChange (v.adicCompletion β„š))) := by let : Algebra β„š (v.adicCompletion β„š) := IsDedekindDomain.HeightOneSpectrum.instAlgebraAdicCompletion (π“ž β„š) β„š v have h := fixedForm_isometric_at_every_finitePlace (F := β„š) (E := β„š) (B₁.restrict (component T p)) (Bβ‚‚.restrict (component U p)) (rat_selfAdjoint _) (rat_selfAdjoint _) (1 : β„š) (by apply top_unique intro x hx exact (Algebra.adjoin β„š ({(1 : β„š)} : Set β„š)).algebraMap_mem x) (fun q hq => componentDatum T U B₁ Bβ‚‚ p (hloc q hq)) v simpa only [fixedForm_rat_canonical] using h lemma restricted_forms_at_every_infinitePlace (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (hreal : RationalPlaceDatum ℝ T U B₁ Bβ‚‚) (v : InfinitePlace β„š) : letI : Algebra β„š v.Completion := NumberField.InfinitePlace.Completion.instAlgebra v β„š Nonempty (((B₁.restrict (component T p)).baseChange v.Completion).IsometryEquiv ((Bβ‚‚.restrict (component U p)).baseChange v.Completion)) := by let : Algebra β„š v.Completion := NumberField.InfinitePlace.Completion.instAlgebra v β„š have h := fixedForm_isometric_at_every_infinitePlace (F := β„š) (E := β„š) (B₁.restrict (component T p)) (Bβ‚‚.restrict (component U p)) (rat_selfAdjoint _) (rat_selfAdjoint _) (1 : β„š) (by apply top_unique intro x hx exact (Algebra.adjoin β„š ({(1 : β„š)} : Set β„š)).algebraMap_mem x) (componentDatum T U B₁ Bβ‚‚ p hreal) v simpa only [fixedForm_rat_canonical] using h lemma diagonal_nondegenerate (B : BilinForm β„š V) (hs : B.IsSymm) (hn : B.Nondegenerate) : B.toQuadraticMap.Nondegenerate := by apply QuadraticMap.nondegenerate_associated_iff.mp rw [QuadraticMap.associated_left_inverse β„š hs.eq] exact hn lemma diagonal_baseChange {K : Type*} [CommRing K] [Algebra β„š K] (B : BilinForm β„š V) : QuadraticForm.baseChange K B.toQuadraticMap = (B.baseChange K).toQuadraticMap := by apply _root_.baseChange_ext intro x simp only [QuadraticForm.baseChange_tmul, LinearMap.BilinMap.toQuadraticMap_apply, LinearMap.BilinForm.baseChange_tmul, mul_one, Algebra.smul_def] def diagonalIsometryAtPlace {K : Type*} [CommRing K] [Algebra β„š K] (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (e : (B₁.baseChange K).IsometryEquiv (Bβ‚‚.baseChange K)) : (QuadraticForm.baseChange K B₁.toQuadraticMap).IsometryEquiv (QuadraticForm.baseChange K Bβ‚‚.toQuadraticMap) where __ := e.toLinearEquiv map_app' x := by rw [diagonal_baseChange, diagonal_baseChange] exact e.map_app x x theorem component_all_places (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (p : β„š[X]) (hreal : RationalPlaceDatum ℝ T U B₁ Bβ‚‚) (hloc : βˆ€ (q : β„•) (_ : Fact q.Prime), RationalPlaceDatum (Padic q) T U B₁ Bβ‚‚) : AllPlacesIsometric (B₁.restrict (component T p)).toQuadraticMap (Bβ‚‚.restrict (component U p)).toQuadraticMap := by constructor Β· intro v let : Algebra β„š (v.adicCompletion β„š) := IsDedekindDomain.HeightOneSpectrum.instAlgebraAdicCompletion (π“ž β„š) β„š v obtain ⟨e⟩ := restricted_forms_at_every_finitePlace T U B₁ Bβ‚‚ p hloc v exact ⟨diagonalIsometryAtPlace _ _ e⟩ Β· intro v let : Algebra β„š v.Completion := NumberField.InfinitePlace.Completion.instAlgebra v β„š obtain ⟨e⟩ := restricted_forms_at_every_infinitePlace T U B₁ Bβ‚‚ p hreal v exact ⟨diagonalIsometryAtPlace _ _ e⟩ theorem finite_order_identity_hasse_input [FiniteDimensional β„š V] [FiniteDimensional β„š W] (T : Module.End β„š V) (U : Module.End β„š W) (B₁ : BilinForm β„š V) (Bβ‚‚ : BilinForm β„š W) (m : β„•) (hm : Odd m) (hTm : T ^ m = 1) (hUm : U ^ m = 1) (hs₁ : B₁.IsSymm) (hsβ‚‚ : Bβ‚‚.IsSymm) (hn₁ : B₁.Nondegenerate) (hnβ‚‚ : Bβ‚‚.Nondegenerate) (hT : βˆ€ x y, B₁ (T x) (T y) = B₁ x y) (hU : βˆ€ x y, Bβ‚‚ (U x) (U y) = Bβ‚‚ x y) (hreal : RationalPlaceDatum ℝ T U B₁ Bβ‚‚) (hloc : βˆ€ (q : β„•) (_ : Fact q.Prime), RationalPlaceDatum (Padic q) T U B₁ Bβ‚‚) : βˆƒ (p : β„š[X]), (X - 1) * p = X ^ m - 1 ∧ IsCoprime (X - 1) p ∧ (B₁.restrict (component T p)).toQuadraticMap.Nondegenerate ∧ (Bβ‚‚.restrict (component U p)).toQuadraticMap.Nondegenerate ∧ AllPlacesIsometric (B₁.restrict (component T p)).toQuadraticMap (Bβ‚‚.restrict (component U p)).toQuadraticMap := by have hmpos : 0 < m := by obtain ⟨k, hk⟩ := hm; omega obtain ⟨p, hp, hc, hfT⟩ := CyclotomicComponentFactor.exists_complement T m 1 hmpos (one_dvd m) hTm rw [cyclotomic_one] at hp hc hfT have hfU : aeval U ((X - 1) * p) = 0 := by rw [hp, map_sub, map_pow, aeval_X, map_one, hUm, sub_self] exact ⟨p, hp, hc, diagonal_nondegenerate (B₁.restrict (component T p)) ⟨fun x y => hs₁.eq x y⟩ (restriction_nondegenerate T B₁ p hfT hc hT hs₁ hn₁), diagonal_nondegenerate (Bβ‚‚.restrict (component U p)) ⟨fun x y => hsβ‚‚.eq x y⟩ (restriction_nondegenerate U Bβ‚‚ p hfU hc hU hsβ‚‚ hnβ‚‚), component_all_places T U B₁ Bβ‚‚ p hreal hloc⟩ end end MultiplicityCounterexample.IdentityComponentAtPlaces end OAI