| import Mathlib |
| import OAI.NumberTheory.PrimePackets.ArtinSeries |
| import OAI.NumberTheory.PrimePackets.PermutationArtin |
|
|
| namespace OAI |
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|
| noncomputable section |
| open Module Filter Topology |
| open scoped NumberField Classical BigOperators |
| namespace PrimeDegreePackets.GaloisIdealCharacter |
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| variable (F E : Type*) [Field F] [NumberField F] [Field E] [NumberField E] |
| variable [Algebra F E] [FiniteDimensional F E] [IsGalois F E] |
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| |
| def rationalPrimeSpectrum (p : Nat.Primes) (P : idealsOverPrime F p.val) : |
| IsDedekindDomain.HeightOneSpectrum (𝓞 F) := by |
| have hP := (mem_idealsOverPrime_iff F p.property.ne_zero P.val).mp P.property |
| refine ⟨P.val, hP.1, ?_⟩ |
| intro he |
| rw [he, Ideal.mem_bot] at hP |
| exact p.property.ne_zero (Nat.cast_eq_zero.mp hP.2) |
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| |
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| def cosetRationalEuler (L : IntermediateField F E) (N : ℕ) (p : Nat.Primes) (s : ℂ) : ℂ := |
| if p.val ∣ N then 1 else |
| ∏ P : idealsOverPrime F p.val, |
| (LinearMap.det (LinearMap.id - (Ideal.absNorm P.val : ℂ)^(-s) • |
| (LinearEquiv.funCongrLeft ℂ ℂ |
| (MulAction.toPerm (β := (E ≃ₐ[F] E) ⧸ L.fixingSubgroup) |
| (primeFrobenius F E (rationalPrimeSpectrum F p P))⁻¹)).toLinearMap))⁻¹ |
|
|
| lemma cosetRationalEuler_good (L : IntermediateField F E) {N : ℕ} |
| (hD : (NumberField.discr E).natAbs ∣ N) (p : Nat.Primes) (hp : ¬ p.val ∣ N) (s : ℂ) : |
| cosetRationalEuler F E L N p s = primeIdealEulerFactor L p.val s := by |
| have hpD : p.val.Coprime (NumberField.discr E).natAbs := |
| (p.property.coprime_iff_not_dvd.mpr hp).of_dvd_right hD |
| let (P : idealsOverPrime F p.val) : P.val.IsPrime := |
| ((mem_idealsOverPrime_iff F p.property.ne_zero P.val).mp P.property).1 |
| let (P : idealsOverPrime F p.val) : Fintype (Ideal.primesOver P.val (𝓞 L)) := |
| (Algebra.QuasiFinite.finite_primesOver P.val).fintype |
| rw [cosetRationalEuler, ite_eq_right hp] |
| unfold primeIdealEulerFactor |
| rw [← (primeIdealTowerEquiv F L p.val p.property.ne_zero).prod_comp] |
| simp only [primeIdealTowerEquiv_val, Fintype.prod_sigma] |
| apply Finset.prod_congr rfl |
| intro P _ |
| let PP := rationalPrimeSpectrum F p P |
| let Q := primeAbove F E PP |
| let : PP.asIdeal.IsMaximal := PP.isPrime.isMaximal PP.ne_bot |
| let : Q.val.IsPrime := Q.property.1 |
| let : Q.val.IsMaximal := Q.property.1.isMaximal (primeAbove_ne_bot F E PP) |
| let : Q.val.LiesOver PP.asIdeal := Q.property.2 |
| let : Algebra.IsUnramifiedAt (𝓞 F) Q.val := |
| unramified_of_absDiscr_not_mem F E PP.asIdeal Q.val Q.property.2.over.symm |
| (Ideal.IsPrime.notMem_of_isCoprime_of_mem hpD.cast |
| ((mem_idealsOverPrime_iff F p.property.ne_zero P.val).mp P.property).2) |
| exact actual_nonnormal_artin_localEuler F E L PP.asIdeal PP.ne_bot Q.val |
| (primeFrobenius F E PP) (primeFrobenius_spec F E PP) s |
|
|
| section EulerCorrection |
| variable (K : Type*) [Field K] [NumberField K] |
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| |
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| def removedEulerPolynomial (N : ℕ) (s : ℂ) : ℂ := |
| ∏ p ∈ badRationalPrimes N, ∏ P : idealsOverPrime K p.val, |
| (1 - (Ideal.absNorm P.val : ℂ)^(-s)) |
|
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| lemma primeEuler_denominator_ne_zero {p : ℕ} (P : idealsOverPrime K p) |
| {s : ℂ} (hs : 0 < s.re) : 1 - (Ideal.absNorm P.val : ℂ)^(-s) ≠ 0 := by |
| have hn : 1 < (Ideal.absNorm P.val : ℝ) := by |
| exact_mod_cast (idealsOverPrime_norm_two_le K P.property) |
| have hnorm : ‖(Ideal.absNorm P.val : ℂ)^(-s)‖ < 1 := by |
| rw [show (Ideal.absNorm P.val : ℂ) = ((Ideal.absNorm P.val : ℝ) : ℂ) by simp, |
| Complex.norm_cpow_eq_rpow_re_of_pos (by positivity)] |
| exact Real.rpow_lt_one_of_one_lt_of_neg hn (by simpa using neg_neg_of_pos hs) |
| exact sub_ne_zero.mpr (fun h => by simp [← h] at hnorm) |
|
|
| lemma removedEulerPolynomial_entire (N : ℕ) : |
| Differentiable ℂ (removedEulerPolynomial K N) := by |
| apply Differentiable.fun_finsetProd |
| intro p _ |
| apply Differentiable.fun_finsetProd |
| intro P _ |
| have hn : (Ideal.absNorm P.val : ℂ) ≠ 0 := by |
| exact_mod_cast (by have := idealsOverPrime_norm_two_le K P.property; omega : |
| Ideal.absNorm P.val ≠ 0) |
| exact (differentiable_const (1 : ℂ)).sub (differentiable_id.neg.const_cpow (Or.inl hn)) |
|
|
| lemma removedEulerPolynomial_ne_zero (N : ℕ) {s : ℂ} (hs : 0 < s.re) : |
| removedEulerPolynomial K N s ≠ 0 := by |
| apply Finset.prod_ne_zero_iff.mpr |
| intro p _ |
| apply Finset.prod_ne_zero_iff.mpr |
| intro P _ |
| exact primeEuler_denominator_ne_zero K P hs |
|
|
| end EulerCorrection |
|
|
| lemma cosetRationalEuler_hasProd (L : IntermediateField F E) (N : ℕ) (hN : 0 < N) |
| (hD : (NumberField.discr E).natAbs ∣ N) {s : ℂ} (hs : 1 < s.re) : |
| HasProd (fun p : Nat.Primes => cosetRationalEuler F E L N p s) |
| (NumberField.dedekindZeta L s * removedEulerPolynomial L N s) := by |
| let C : Nat.Primes → ℂ := fun p => |
| if p ∈ badRationalPrimes N then |
| ∏ P : idealsOverPrime L p.val, (1 - (Ideal.absNorm P.val : ℂ)^(-s)) else 1 |
| have hC : HasProd C (removedEulerPolynomial L N s) := by |
| have h : HasProd C (∏ p ∈ badRationalPrimes N, C p) := hasProd_prod_of_ne_finset_one (s := badRationalPrimes N) (f := C) |
| (fun p hp => ite_eq_right hp) |
| simpa only [C, Finset.prod_ite_mem, Finset.inter_self, removedEulerPolynomial] using h |
| apply ((dedekindZeta_hasProd L hs).mul hC).congr_fun |
| intro p |
| have hpN : p ∈ badRationalPrimes N ↔ p.val ∣ N := by |
| change (⟨p.val, p.property⟩ : Subtype Nat.Prime) ∈ N.primeFactors.subtype Nat.Prime ↔ _ |
| rw [Finset.mem_subtype, Nat.mem_primeFactors] |
| exact ⟨fun h => h.2.1, fun h => ⟨p.property, h, hN.ne'⟩⟩ |
| by_cases hp : p.val ∣ N |
| · simp only [cosetRationalEuler, ite_eq_left hp, C, ite_eq_left (hpN.mpr hp)] |
| unfold primeIdealEulerFactor |
| rw [← Finset.prod_mul_distrib] |
| simp only [inv_mul_cancel₀ (primeEuler_denominator_ne_zero L _ (zero_lt_one.trans hs)), |
| Finset.prod_const_one] |
| · simp only [C, ite_eq_right (mt hpN.mp hp), mul_one] |
| exact cosetRationalEuler_good F E L hD p hp s |
|
|
| open HeckeMellin |
| section ContinuedCoset |
| variable {d : ℕ} {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyReal K] |
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| |
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| def cosetMeromorphic (σ : OrderedEmbeddings (d+1) K) (hn : finrank ℚ K = d+1) |
| (N : ℕ) (s : ℂ) : ℂ := |
| removedEulerPolynomial K N s * (regularizedThetaZeta σ hn s / (s-1)) |
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| def cosetXiFactor (N : ℕ) (s : ℂ) : ℂ := |
| removedEulerPolynomial K N s * |
| (completionExponential (d+1) |(NumberField.discr K : ℝ)| s)⁻¹ * |
| ((Complex.Gamma (s/2))⁻¹)^(d+1) / s / (s-1) |
|
|
| lemma gammaInverse_entire : |
| Differentiable ℂ (fun s : ℂ => (Complex.Gamma (s/2))⁻¹) := by |
| simpa only [one_div, Function.comp_def, id_eq] using |
| (Complex.differentiable_one_div_Gamma.comp (differentiable_id.div_const 2)) |
|
|
| omit [NumberField.IsTotallyReal K] in |
| lemma exponentialInverse_entire : Differentiable ℂ |
| (fun s : ℂ => (completionExponential (d+1) |(NumberField.discr K : ℝ)| s)⁻¹) := by |
| have HE : Differentiable ℂ (completionExponential (d+1) |(NumberField.discr K : ℝ)|) := |
| fun s => (completionExponential_hasDerivAt _ _ s).differentiableAt |
| exact HE.inv (fun s => completionExponential_ne_zero _ _ s) |
|
|
| omit [NumberField.IsTotallyReal K] in |
| lemma cosetXiFactor_meromorphicAt (N : ℕ) (s : ℂ) : |
| MeromorphicAt (cosetXiFactor (K := K) (d := d) N) s := by |
| have hA := (((removedEulerPolynomial_entire K N).analyticAt s).mul |
| ((exponentialInverse_entire (K := K) (d := d)).analyticAt s)).mul |
| ((gammaInverse_entire.analyticAt s).pow (d+1)) |
| exact (hA.meromorphicAt.div analyticAt_id.meromorphicAt).div |
| (analyticAt_id.sub analyticAt_const).meromorphicAt |
|
|
| omit [NumberField.IsTotallyReal K] in |
| lemma cosetXiFactor_analyticAt (N : ℕ) {s : ℂ} (hs0 : s ≠ 0) (hs1 : s ≠ 1) : |
| AnalyticAt ℂ (cosetXiFactor (K := K) (d := d) N) s := by |
| have hA := (((removedEulerPolynomial_entire K N).analyticAt s).mul |
| ((exponentialInverse_entire (K := K) (d := d)).analyticAt s)).mul |
| ((gammaInverse_entire.analyticAt s).pow (d+1)) |
| exact (hA.div analyticAt_id hs0).div (analyticAt_id.sub analyticAt_const) |
| (sub_ne_zero.mpr hs1) |
|
|
| omit [NumberField.IsTotallyReal K] in |
| lemma cosetXiFactor_ne_zero (N : ℕ) {s : ℂ} (hs : 0 < s.re) (hs1 : s ≠ 1) : |
| cosetXiFactor (K := K) (d := d) N s ≠ 0 := by |
| apply div_ne_zero _ (sub_ne_zero.mpr hs1) |
| apply div_ne_zero _ (by contrapose! hs; simp [hs]) |
| apply mul_ne_zero |
| · exact mul_ne_zero (removedEulerPolynomial_ne_zero K N hs) |
| (inv_ne_zero (completionExponential_ne_zero _ _ s)) |
| · exact pow_ne_zero _ (inv_ne_zero (Complex.Gamma_ne_zero (half_ne_nonpositive_integer hs))) |
| |
| lemma cosetMeromorphic_eq_factor (σ : OrderedEmbeddings (d+1) K) |
| (hn : finrank ℚ K = d+1) (N : ℕ) : |
| cosetMeromorphic σ hn N = cosetXiFactor (K := K) (d := d) N * completedThetaXi σ hn := by |
| funext s |
| simp only [cosetMeromorphic, cosetXiFactor, regularizedThetaZeta, Pi.mul_apply, |
| div_eq_mul_inv] |
| ring |
|
|
| lemma cosetMeromorphic_meromorphicAt (σ : OrderedEmbeddings (d+1) K) |
| (hn : finrank ℚ K = d+1) (N : ℕ) (s : ℂ) : MeromorphicAt (cosetMeromorphic σ hn N) s := by |
| rw [cosetMeromorphic_eq_factor] |
| exact (cosetXiFactor_meromorphicAt N s).mul |
| ((completedThetaXi_entire σ hn).analyticAt s).meromorphicAt |
|
|
| lemma cosetMeromorphic_eq_dedekindZeta (σ : OrderedEmbeddings (d+1) K) |
| (hn : finrank ℚ K = d+1) (N : ℕ) {s : ℂ} (hs : 1 < s.re) : |
| cosetMeromorphic σ hn N s = NumberField.dedekindZeta K s * removedEulerPolynomial K N s := by |
| have hs1 : s - 1 ≠ 0 := by |
| apply sub_ne_zero.mpr |
| intro he |
| simp [he] at hs |
| rw [cosetMeromorphic, regularizedThetaZeta_eq_totallyReal σ hn hs, |
| mul_div_cancel_left₀ _ hs1, mul_comm] |
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| |
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| theorem cosetMeromorphic_order_eq (σ : OrderedEmbeddings (d+1) K) |
| (hn : finrank ℚ K = d+1) (N : ℕ) {β : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) : |
| meromorphicOrderAt (cosetMeromorphic σ hn N) (β : ℂ) = |
| meromorphicOrderAt (completedThetaXi σ hn) (β : ℂ) := by |
| have h0 : (β : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hβ0.ne' |
| have h1 : (β : ℂ) ≠ 1 := by exact_mod_cast hβ1.ne |
| rw [cosetMeromorphic_eq_factor] |
| exact meromorphicOrderAt_mul_of_ne_zero (cosetXiFactor_analyticAt N h0 h1) |
| (cosetXiFactor_ne_zero N hβ0 h1) |
|
|
| end ContinuedCoset |
|
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| |
|
|
| theorem actual_coset_meromorphic_pairing |
| (L : IntermediateField F E) [NumberField.IsTotallyReal L] {d : ℕ} |
| (σ : OrderedEmbeddings (d+1) L) (hn : finrank ℚ L = d+1) |
| (N : ℕ) (hN : 0 < N) (hD : (NumberField.discr E).natAbs ∣ N) : |
| (∀ s : ℂ, MeromorphicAt (cosetMeromorphic σ hn N) s) ∧ |
| (∀ s : ℂ, 1 < s.re → HasProd (fun p : Nat.Primes => cosetRationalEuler F E L N p s) |
| (cosetMeromorphic σ hn N s)) ∧ |
| (∀ β : ℝ, 0 < β → β < 1 → |
| meromorphicOrderAt (cosetMeromorphic σ hn N) (β : ℂ) = |
| meromorphicOrderAt (completedThetaXi σ hn) (β : ℂ)) := by |
| refine ⟨cosetMeromorphic_meromorphicAt σ hn N, ?_, ?_⟩ |
| · intro s hs |
| rw [cosetMeromorphic_eq_dedekindZeta σ hn N hs] |
| exact cosetRationalEuler_hasProd F E L N hN hD hs |
| · intro β hβ0 hβ1 |
| exact cosetMeromorphic_order_eq σ hn N hβ0 hβ1 |
|
|
| end PrimeDegreePackets.GaloisIdealCharacter |
|
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| end |
|
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| end OAI |
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