| import Mathlib |
| import OAI.NumberTheory.CubicGram.CubeExtraction |
| import OAI.NumberTheory.CubicGram.LatticeCounts |
|
|
| namespace OAI |
|
|
| /-! Ramified residual factors and full cube decomposition. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
|
| theorem cube_decomposition (n : Eisenstein) : |
| ∃ s t c : Eisenstein, Squarefree s ∧ Squarefree t ∧ IsCoprime s t ∧ n = s*t^2*c^3 := by |
| refine UniqueFactorizationMonoid.induction_on_prime n ?_ ?_ ?_ |
| · exact ⟨1,1,0,squarefree_one,squarefree_one,isCoprime_one_left,by simp⟩ |
| · intro u hu |
| exact ⟨u,1,1,hu.squarefree,squarefree_one,isCoprime_one_right,by simp⟩ |
| · intro b p hb hp ih |
| obtain ⟨s,t,c,hss,hst,hcop,rfl⟩ := ih |
| by_cases hps : p ∣ s |
| · obtain ⟨s',rfl⟩ := hps |
| have hps' : IsCoprime p s' := isRelPrime_iff_isCoprime.mp (squarefree_mul_iff.mp hss).1 |
| have hpt : IsCoprime p t := hcop.of_mul_left_left |
| refine ⟨s',p*t,c,hss.of_mul_right, |
| squarefree_mul_iff.mpr ⟨hpt.isRelPrime,hp.squarefree,hst⟩, |
| hps'.symm.mul_right hcop.of_mul_left_right,?_⟩ |
| ring |
| · by_cases hpt : p ∣ t |
| · obtain ⟨t',rfl⟩ := hpt |
| refine ⟨s,t',p*c,hss,hst.of_mul_right,hcop.of_mul_right_right,?_⟩ |
| ring |
| · have hps' := hp.coprime_iff_not_dvd.mpr hps |
| have hpt' := hp.coprime_iff_not_dvd.mpr hpt |
| refine ⟨p*s,t,c,squarefree_mul_iff.mpr ⟨hps'.isRelPrime,hp.squarefree,hss⟩, |
| hst,hpt'.mul_left hcop,?_⟩ |
| ring |
|
|
| theorem squarefree_ramified_split (n : Eisenstein) (hs : Squarefree n) : |
| ∃ d e : Eisenstein, d ∣ 3 ∧ IsCoprime e 3 ∧ n = d*e := by |
| have h : ∀ n : Eisenstein, Squarefree n → |
| ∃ d e : Eisenstein, d ∣ 3 ∧ IsCoprime e 3 ∧ n = d*e := by |
| intro n |
| refine UniqueFactorizationMonoid.induction_on_prime n ?_ ?_ ?_ |
| · intro hs |
| have hh := hs 0 (by simp) |
| exact (not_isUnit_zero hh).elim |
| · intro u hu hs |
| refine ⟨1,u,one_dvd _,?_,by simp⟩ |
| exact (isCoprime_zero_right.mpr hu).of_isCoprime_of_dvd_right (dvd_zero 3) |
| · intro b p hb hp ih hs |
| obtain ⟨d,e,hd,he,hbe⟩ := ih hs.of_mul_right |
| have hpb : IsCoprime p b := isRelPrime_iff_isCoprime.mp (squarefree_mul_iff.mp hs).1 |
| by_cases hp3 : p ∣ 3 |
| · have hpd : IsCoprime p d := (hbe ▸ hpb).of_mul_right_left |
| refine ⟨p*d,e,hpd.mul_dvd hp3 hd,he,?_⟩ |
| rw [hbe]; ring |
| · refine ⟨d,p*e,hd,(hp.coprime_iff_not_dvd.mpr hp3).mul_left he,?_⟩ |
| rw [hbe]; ring |
| exact h n hs |
|
|
| lemma unit_residue_of_isCoprime_three {e : Eisenstein} (he : IsCoprime e 3) : |
| IsUnit (Ideal.Quotient.mk (modulus 3) e) := by |
| have h := he.map (Ideal.Quotient.mk (modulus 3)) |
| simpa only [residue_three_eq_zero, isCoprime_zero_right] using h |
|
|
| lemma norm_le_of_dvd {a b : Eisenstein} (hb : b ≠ 0) (hd : a ∣ b) : |
| norm a ≤ norm b := by |
| obtain ⟨c,rfl⟩ := hd |
| have hc : c ≠ 0 := by intro h; simp [h] at hb |
| rw [norm_mul_eq] |
| exact le_mul_of_one_le_right (norm_nonneg a) (one_le_norm hc) |
|
|
| theorem squarefree_primary_residual (n : Eisenstein) (hs : Squarefree n) : |
| ∃ r s : Eisenstein, r ≠ 0 ∧ norm r ≤ 9 ∧ primary s ∧ Squarefree s ∧ n = r*s := by |
| obtain ⟨d,e,hd,he,hde⟩ := squarefree_ramified_split n hs |
| obtain ⟨u,hu,hprim⟩ := unit_lift_mod_three (unit_residue_of_isCoprime_three he) |
| obtain ⟨v,rfl⟩ := hu |
| have hd0 : d ≠ 0 := by |
| intro hd0 |
| rw [hd0, zero_dvd_iff] at hd |
| norm_num at hd |
| have hvi : norm (↑v⁻¹ : Eisenstein) = 1 := by |
| rw [← normNat_cast, normNat_of_isUnit (Units.isUnit _), Nat.cast_one] |
| refine ⟨d*(↑v⁻¹ : Eisenstein),e*(↑v : Eisenstein),mul_ne_zero hd0 (Units.ne_zero _),?_,hprim,?_,?_⟩ |
| · rw [norm_mul_eq, hvi, mul_one] |
| have h := norm_le_of_dvd (by norm_num : (3 : Eisenstein) ≠ 0) hd |
| have h3 : norm (3 : Eisenstein) = 9 := by |
| change Complex.normSq (3 : ℂ) = 9 |
| norm_num |
| simpa only [h3] using h |
| · have hes : Squarefree e := (hde ▸ hs).of_mul_right |
| exact squarefree_mul_iff.mpr ⟨((isCoprime_zero_left.mpr (Units.isUnit v)).of_isCoprime_of_dvd_left (dvd_zero e)).isRelPrime,hes, |
| (Units.isUnit v).squarefree⟩ |
| · rw [hde] |
| simp [mul_assoc, mul_left_comm, mul_comm] |
|
|
| end CubicFirstMoment |
|
|
| namespace CubicFirstMoment |
|
|
| theorem full_cube_decomposition {n : Eisenstein} (hn : n ≠ 0) : |
| ∃ r s t c : Eisenstein, r ≠ 0 ∧ norm r ≤ 729 ∧ |
| primary s ∧ primary t ∧ Squarefree s ∧ Squarefree t ∧ IsCoprime s t ∧ |
| c ≠ 0 ∧ n = r*s*t^2*c^3 := by |
| obtain ⟨s₀,t₀,c,hs₀,ht₀,hst,hn'⟩ := cube_decomposition n |
| obtain ⟨r,s,hr0,hr,hs,hss,hs'⟩ := squarefree_primary_residual s₀ hs₀ |
| obtain ⟨q,t,hq0,hq,ht,hst',ht'⟩ := squarefree_primary_residual t₀ ht₀ |
| have hc0 : c ≠ 0 := by intro hc; simp [hc] at hn'; exact hn hn' |
| have hcop : IsCoprime s t := |
| hst.of_isCoprime_of_dvd_left (hs' ▸ dvd_mul_left s r) |>.of_isCoprime_of_dvd_right |
| (ht' ▸ dvd_mul_left t q) |
| refine ⟨r*q^2,s,t,c,mul_ne_zero hr0 (pow_ne_zero _ hq0),?_,hs,ht,hss,hst',hcop,hc0,?_⟩ |
| · rw [norm_mul_eq] |
| have he : norm (q^2) = norm q ^ 2 := by simp [pow_two] |
| rw [he] |
| have hq2 : norm q ^ 2 ≤ 81 := by nlinarith [norm_nonneg q] |
| nlinarith [norm_nonneg r, sq_nonneg (norm q)] |
| · rw [hn',hs',ht']; ring |
|
|
| def residualCubeSupport (R S T C : Finset Eisenstein) : Finset Eisenstein := |
| R.biUnion fun r => (cubeBlockSupport S T C).image (r * ·) |
|
|
| lemma residualCubeSupport_mass_le (P R S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) (u : Eisenstein → ℂ) : |
| (∑ n ∈ residualCubeSupport R S T C, ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) ≤ |
| (R.card : ℝ) * |
| min ((T.card : ℝ)*C.card*finiteCubicBound P S) |
| ((S.card : ℝ)*C.card*finiteCubicBound P T) * ∑ b ∈ P, ‖u b‖^2 := by |
| let L := min ((T.card : ℝ)*C.card*finiteCubicBound P S) |
| ((S.card : ℝ)*C.card*finiteCubicBound P T) |
| have htw (r : Eisenstein) : |
| (∑ n ∈ (cubeBlockSupport S T C).image (r * ·), ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) ≤ |
| L * ∑ b ∈ P, ‖u b‖^2 := by |
| have h := finiteCubicBound_twist_controls_mass P (cubeBlockSupport S T C) u |
| (fun b => cubicSymbol b r) (fun b hb => norm_cubicSymbol_le_one (hP b hb) r) |
| calc |
| _ ≤ ∑ n ∈ cubeBlockSupport S T C, ‖∑ b ∈ P, u b*cubicSymbol b (r*n)‖^2 := |
| Finset.sum_image_le_of_nonneg (g := fun n => r*n) |
| (f := fun n => ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) (fun _ _ => sq_nonneg _) |
| _ = ∑ n ∈ cubeBlockSupport S T C, ‖∑ b ∈ P, u b*cubicSymbol b r*cubicSymbol b n‖^2 := by |
| apply Finset.sum_congr rfl; intro n hn |
| congr 2 |
| apply Finset.sum_congr rfl; intro b hb |
| rw [cubicSymbol_mul_upper (hP b hb)]; ring |
| _ ≤ finiteCubicBound P (cubeBlockSupport S T C) * ∑ b ∈ P, ‖u b‖^2 := h |
| _ ≤ _ := mul_le_mul_of_nonneg_right (finiteCubicBound_cubeBlock P S T C hP) |
| (Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
| have hunion : (∑ n ∈ residualCubeSupport R S T C, ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) ≤ |
| ∑ r ∈ R, ∑ n ∈ (cubeBlockSupport S T C).image (r * ·), |
| ‖∑ b ∈ P, u b*cubicSymbol b n‖^2 := by |
| classical |
| induction R using Finset.induction_on with |
| | empty => simp [residualCubeSupport] |
| | @insert r R hr ih => |
| rw [residualCubeSupport, Finset.biUnion_insert, Finset.sum_insert hr] |
| dsimp only [residualCubeSupport] at ih |
| have hh : ∀ n ∈ (cubeBlockSupport S T C).image (r * ·) ∩ |
| R.biUnion (fun r => (cubeBlockSupport S T C).image (r * ·)), |
| (0 : ℝ) ≤ ‖∑ b ∈ P, u b*cubicSymbol b n‖^2 := fun _ _ => sq_nonneg _ |
| have he := Finset.sum_union_inter (s₁ := (cubeBlockSupport S T C).image (r * ·)) |
| (s₂ := R.biUnion (fun r => (cubeBlockSupport S T C).image (r * ·))) |
| (f := fun n => ‖∑ b ∈ P, u b*cubicSymbol b n‖^2) |
| have hn := Finset.sum_nonneg hh |
| linarith |
| calc |
| _ ≤ ∑ r ∈ R, ∑ n ∈ (cubeBlockSupport S T C).image (r * ·), |
| ‖∑ b ∈ P, u b*cubicSymbol b n‖^2 := hunion |
| _ ≤ ∑ r ∈ R, L * ∑ b ∈ P, ‖u b‖^2 := Finset.sum_le_sum (fun r _ => htw r) |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]; ring |
|
|
| theorem finiteCubicBound_residualCube (P R S T C : Finset Eisenstein) |
| (hP : ∀ b ∈ P, primary b) : |
| finiteCubicBound P (residualCubeSupport R S T C) ≤ |
| (R.card : ℝ) * min ((T.card : ℝ)*C.card*finiteCubicBound P S) |
| ((S.card : ℝ)*C.card*finiteCubicBound P T) := by |
| apply (finiteCubicBound_le_iff _ _ (mul_nonneg (by positivity) |
| (le_min (by positivity [finiteCubicBound_nonneg P S]) |
| (by positivity [finiteCubicBound_nonneg P T])))).mpr |
| exact residualCubeSupport_mass_le P R S T C hP |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| end OAI |
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|
