| import OAI.AlgebraicGeometry.PlaneCurves.SectionOrder |
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| namespace OAI |
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| /-! |
| # Coefficient and polynomial transforms for marked normal polynomials |
| -/ |
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| section |
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| namespace Nagata.CoefficientSpaces |
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| / |
| identity valid even at the zeros of Pi. -/ |
| theorem exists_transformed_coefficient {τ L Q : ℂ} |
| (hτ : τ ≠ 0) (hL : L ≠ 0) (hQ : Q ≠ 0) (d : ℤ) (m j : ℕ) |
| (P : Nagata.W08.automorphicSections τ 9 Q) |
| (f : Nagata.W08.automorphicSections τ (3 * (d - 3 * (j : ℤ))) |
| (L ^ (d - 3 * (j : ℤ)))) |
| (hzero : j ≤ m → ∀ z ≠ 0, P.val z = 0 → deriv P.val z ≠ 0 ∧ |
| ((m - j : ℕ) : ℕ∞) ≤ analyticOrderAt f.val z) : |
| ∃ g ∈ coefficientSpace τ L Q d (m : ℤ) (j : ℤ) P.val, |
| ∀ z ≠ 0, P.val z ^ m * g z = P.val z ^ j * f.val z := by |
| by_cases hj : j ≤ m |
| · have hjZ : (j : ℤ) ≤ (m : ℤ) := by exact_mod_cast hj |
| have he : ((m : ℤ) - (j : ℤ)).toNat = m - j := by omega |
| obtain ⟨g, hg, hf⟩ := exists_low_holomorphic_coefficient hτ hL hQ d (m : ℤ) |
| (j : ℤ) hjZ P f (by simpa only [he] using hzero hj) |
| refine ⟨g, hg, ?_⟩ |
| intro z hz |
| rw [hf z hz, he, ← mul_assoc, ← pow_add] |
| rw [Nat.add_sub_of_le hj] |
| · have hjZ : (m : ℤ) < (j : ℤ) := by exact_mod_cast (lt_of_not_ge hj) |
| have he : ((j : ℤ) - (m : ℤ)).toNat = j - m := by omega |
| refine ⟨multiplyPower P.val (j - m) f.val, ?_, ?_⟩ |
| · apply (mem_coefficientSpace_second hjZ).mpr |
| exact ⟨f.val, f.property, by rw [he]⟩ |
| · intro z hz |
| change P.val z ^ m * (P.val z ^ (j - m) * f.val z) = P.val z ^ j * f.val z |
| rw [← mul_assoc, ← pow_add, Nat.add_sub_of_le (le_of_lt (lt_of_not_ge hj))] |
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| / |
| section into the zero coefficient. -/ |
| theorem transformed_coefficient_nonzero {τ L Q : ℂ} (d : ℤ) (m j : ℕ) |
| (P : Nagata.W08.automorphicSections τ 9 Q) |
| (hPfin : ∀ z ≠ 0, analyticOrderAt P.val z ≠ ⊤) |
| (f : Nagata.W08.automorphicSections τ (3 * (d - 3 * (j : ℤ))) |
| (L ^ (d - 3 * (j : ℤ)))) (g : ℂ → ℂ) |
| (hfactor : ∀ z ≠ 0, P.val z ^ m * g z = P.val z ^ j * f.val z) |
| (hf : f ≠ 0) : g ≠ 0 := by |
| intro hg |
| apply hf |
| apply multiplySection_injective τ (L ^ (d - 3 * (j : ℤ))) |
| (3 * (d - 3 * (j : ℤ))) P.val j |
| (Nagata.W08.automorphicSection_analyticOnNhd P) hPfin |
| funext z |
| change P.val z ^ j * f.val z = P.val z ^ j * 0 |
| by_cases hz : z = 0 |
| · subst z |
| rw [f.property.1] |
| · rw [← hfactor z hz, hg] |
| simp |
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| end Nagata.CoefficientSpaces |
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| end |
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| section |
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| namespace Nagata.CoefficientSpaces |
| open scoped BigOperators |
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| theorem exists_transformed_polynomial {τ L Q : ℂ} |
| (hτ : τ ≠ 0) (hL : L ≠ 0) (hQ : Q ≠ 0) (d : ℤ) (m : ℕ) |
| (P : Nagata.W08.automorphicSections τ 9 Q) |
| (hPsimple : ∀ z ≠ 0, P.val z = 0 → deriv P.val z ≠ 0) |
| (F : Polynomial (ℂ → ℂ)) |
| (hbound : ∀ j : ℕ, (d / 3).toNat < j → F.coeff j = 0) |
| (hsource : ∀ j : ℕ, F.coeff j ∈ Nagata.W08.automorphicSections τ |
| (3 * (d - 3 * (j : ℤ))) (L ^ (d - 3 * (j : ℤ)))) |
| (hzero : ∀ j : ℕ, j ≤ (d / 3).toNat → j ≤ m → ∀ z ≠ 0, |
| P.val z = 0 → ((m - j : ℕ) : ℕ∞) ≤ analyticOrderAt (F.coeff j) z) : |
| ∃ G : polynomialSections τ L Q d (m : ℤ) P.val, |
| (∀ j : ℕ, ∀ z ≠ 0, P.val z ^ m * G.val.coeff j z = |
| P.val z ^ j * F.coeff j z) ∧ (F ≠ 0 → G ≠ 0) := by |
| classical |
| let N := (d / 3).toNat |
| have hex : ∀ j : Fin (N + 1), ∃ g ∈ coefficientSpace τ L Q d (m : ℤ) (j.val : ℤ) P.val, |
| ∀ z ≠ 0, P.val z ^ m * g z = P.val z ^ j.val * F.coeff j.val z := by |
| intro j |
| exact exists_transformed_coefficient hτ hL hQ d m j.val P |
| ⟨F.coeff j.val, hsource j.val⟩ (fun hj z hz hPz => |
| ⟨hPsimple z hz hPz, hzero j.val (Nat.le_of_lt_succ j.isLt) hj z hz hPz⟩) |
| choose g hg hfactor using hex |
| let c : (j : Fin (N + 1)) → coefficientSpace τ L Q d (m : ℤ) (j.val : ℤ) P.val := |
| fun j => ⟨g j, hg j⟩ |
| let e := Nagata.Workers.W10.polynomialSectionCoordinates τ L Q d (m : ℤ) P.val |
| let G := e.symm c |
| have hcoeff (j : Fin (N + 1)) : G.val.coeff j.val = g j := by |
| have hh := congrArg (fun v => (v j).val) (e.apply_symm_apply c) |
| exact hh |
| have hfact : ∀ j : ℕ, ∀ z ≠ 0, P.val z ^ m * G.val.coeff j z = |
| P.val z ^ j * F.coeff j z := by |
| intro j z hz |
| by_cases hj : j ≤ N |
| · let ji : Fin (N + 1) := ⟨j, Nat.lt_succ_of_le hj⟩ |
| rw [hcoeff ji] |
| exact hfactor ji z hz |
| · have hj' : N < j := lt_of_not_ge hj |
| rw [G.property.1 j hj', hbound j hj'] |
| simp |
| refine ⟨G, hfact, ?_⟩ |
| intro hF hG |
| apply hF |
| apply Polynomial.ext |
| intro j |
| by_cases hj : j ≤ N |
| · by_contra hfj |
| let f : Nagata.W08.automorphicSections τ (3 * (d - 3 * (j : ℤ))) |
| (L ^ (d - 3 * (j : ℤ))) := ⟨F.coeff j, hsource j⟩ |
| have hf : f ≠ 0 := by |
| intro hf0 |
| apply hfj |
| exact congrArg Subtype.val hf0 |
| have hPfin : ∀ z ≠ 0, analyticOrderAt P.val z ≠ ⊤ := by |
| intro z hz |
| have hp := Nagata.W08.automorphicSection_analyticAt P hz |
| by_cases hPz : P.val z = 0 |
| · rw [hp.analyticOrderAt_eq_one_of_zero_deriv_ne_zero hPz (hPsimple z hz hPz)] |
| simp |
| · rw [hp.analyticOrderAt_eq_zero.mpr hPz] |
| simp |
| have hg := transformed_coefficient_nonzero d m j P hPfin f (G.val.coeff j) |
| (hfact j) hf |
| apply hg |
| have hGval := congrArg Subtype.val hG |
| rw [hGval] |
| rfl |
| · exact hbound j (lt_of_not_ge hj) |
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|
| / |
| theorem scalarExpression_eq_range (F : Polynomial (ℂ → ℂ)) (N : ℕ) |
| (hbound : ∀ j, N < j → F.coeff j = 0) (z v : ℂ) : |
| scalarExpression F z v = ∑ j ∈ Finset.range (N + 1), F.coeff j z * v ^ j := by |
| apply Polynomial.sum_eq_of_subset |
| · intro j |
| simp |
| · intro j hj |
| apply Finset.mem_range.mpr |
| have hne := Polynomial.mem_support_iff.mp hj |
| by_contra h |
| exact hne (hbound j (by omega)) |
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|
| / |
| substitution identity, not merely a matrix or an abstract coefficient model. -/ |
| theorem scalarExpression_transform (F G : Polynomial (ℂ → ℂ)) (N m : ℕ) |
| (hF : ∀ j, N < j → F.coeff j = 0) (hG : ∀ j, N < j → G.coeff j = 0) |
| (P : ℂ → ℂ) (z v : ℂ) |
| (hfactor : ∀ j, P z ^ m * G.coeff j z = P z ^ j * F.coeff j z) : |
| P z ^ m * scalarExpression G z v = scalarExpression F z (P z * v) := by |
| rw [scalarExpression_eq_range G N hG, scalarExpression_eq_range F N hF, |
| Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j hj |
| rw [← mul_assoc, hfactor j, mul_pow] |
| ring |
|
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| end Nagata.CoefficientSpaces |
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| end |
|
|
| section |
|
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| namespace Nagata.CoefficientSpaces |
| open scoped BigOperators |
|
|
| / |
| noncomputable def nineThetaSection {τ : ℂ} (hτ : ‖τ‖ < 1) (hτ0 : τ ≠ 0) |
| (a : Fin 9 → ℂ) (ha : ∀ i, a i ≠ 0) : |
| Nagata.W08.automorphicSections τ 9 (∏ i, -a i) := by |
| simpa only [Fintype.card_fin, Nat.cast_ofNat] using |
| Nagata.W08.markedThetaSection hτ hτ0 a ha |
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|
| / |
| orders to the complete divisor, and the genuine theta product supplies simplicity. -/ |
| theorem exists_marked_transformed_polynomial {τ L : ℂ} |
| (hτ : ‖τ‖ < 1) (hτ0 : τ ≠ 0) (hL : L ≠ 0) |
| (a : Fin 9 → ℂ) (ha : ∀ i, a i ≠ 0) |
| (hdisjoint : Pairwise (fun i j => ∀ k : ℤ, a i ≠ a j * τ ^ k)) |
| (d : ℤ) (m : ℕ) (F : Polynomial (ℂ → ℂ)) |
| (hbound : ∀ j : ℕ, (d / 3).toNat < j → F.coeff j = 0) |
| (hsource : ∀ j : ℕ, F.coeff j ∈ Nagata.W08.automorphicSections τ |
| (3 * (d - 3 * (j : ℤ))) (L ^ (d - 3 * (j : ℤ)))) |
| (hmarked : ∀ j : ℕ, j ≤ (d / 3).toNat → j ≤ m → ∀ i : Fin 9, |
| ((m - j : ℕ) : ℕ∞) ≤ analyticOrderAt (F.coeff j) (a i)) : |
| ∃ G : polynomialSections τ L (∏ i, -a i) d (m : ℤ) |
| (nineThetaSection hτ hτ0 a ha).val, |
| (∀ j : ℕ, ∀ z ≠ 0, (nineThetaSection hτ hτ0 a ha).val z ^ m * |
| G.val.coeff j z = (nineThetaSection hτ hτ0 a ha).val z ^ j * F.coeff j z) ∧ |
| (F ≠ 0 → G ≠ 0) := by |
| have hQ : (∏ i, -a i) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i _ => neg_ne_zero.mpr (ha i)) |
| refine exists_transformed_polynomial hτ0 hL hQ d m |
| (nineThetaSection hτ hτ0 a ha) ?_ F hbound hsource ?_ |
| · intro z hz hzero |
| exact Nagata.W08.markedThetaSection_zero_has_nonzero_deriv hτ hτ0 a ha hdisjoint hz hzero |
| · intro j hj hjm z hz hzero |
| obtain ⟨i, k, rfl⟩ := (Nagata.W08.markedThetaSection_zero_iff hτ hτ0 a ha hz).mp hzero |
| rw [automorphic_order_orbit hτ0 (zpow_ne_zero _ hL) |
| (⟨F.coeff j, hsource j⟩ : Nagata.W08.automorphicSections τ |
| (3 * (d - 3 * (j : ℤ))) (L ^ (d - 3 * (j : ℤ)))) (ha i) k] |
| exact hmarked j hj hjm i |
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| end Nagata.CoefficientSpaces |
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| end |
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| end OAI |
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