| import OAI.NumberTheory.SingleFold.DescentMap |
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| namespace OAI |
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| namespace SingleFold.SquareSupport |
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| lemma squarefree_decomposition {x : ℚ} (hx : x ≠ 0) : |
| ∃ (d : ℕ) (r : ℚ), 0 < d ∧ r ≠ 0 ∧ Squarefree d ∧ |x| = d*r^2 := by |
| have hn : 0 < x.num.natAbs := Int.natAbs_pos.mpr (Rat.num_ne_zero.mpr hx) |
| obtain ⟨d,k,hd,hk,he,hf⟩ := Nat.sq_mul_squarefree_of_pos (Nat.mul_pos hn x.den_pos) |
| refine ⟨d,(k:ℚ)/x.den,hd,div_ne_zero (by exact_mod_cast hk.ne') (by exact_mod_cast x.den_ne_zero),hf,?_⟩ |
| have he' : (k:ℚ)^2*d=x.num.natAbs*x.den := by exact_mod_cast he |
| rw [Rat.abs_def, Rat.divInt_eq_div] |
| push_cast |
| field_simp |
| have habs : |(x.num:ℚ)| = (x.num.natAbs:ℚ) := by simp |
| rw [habs] |
| nlinarith only [he'] |
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|
| lemma val_add_eq {p : ℕ} [Fact p.Prime] {x y : ℚ} |
| (hx : x ≠ 0) (hy : y ≠ 0) (hval : padicValRat p x ≠ padicValRat p y) : |
| padicValRat p (x+y) = min (padicValRat p x) (padicValRat p y) := by |
| apply padicValRat.add_eq_min _ hx hy hval |
| intro h |
| apply hval |
| have he : x = -y := by linarith |
| simp [he] |
|
|
| lemma even_val_triple {p : ℕ} [Fact p.Prime] {x y a b : ℚ} |
| (hx : x ≠ 0) (hxa : x+a ≠ 0) (hxb : x+b ≠ 0) |
| (ha : a ≠ 0) (hb : b ≠ 0) |
| (hva : padicValRat p a = 0) (hvb : padicValRat p b = 0) |
| (he : y^2=x*(x+a)*(x+b)) : Even (padicValRat p x) := by |
| have hv := congrArg (padicValRat p) he |
| rw [padicValRat.pow, padicValRat.mul (mul_ne_zero hx hxa) hxb, |
| padicValRat.mul hx hxa] at hv |
| by_cases hz : padicValRat p x=0 |
| · simp [hz] |
| have h₁ := val_add_eq (p := p) hx ha (by simpa [hva] using hz) |
| have h₂ := val_add_eq (p := p) hx hb (by simpa [hvb] using hz) |
| rw [hva] at h₁ |
| rw [hvb] at h₂ |
| rcases lt_or_gt_of_ne hz with hl | hl |
| · rw [min_eq_left hl.le] at h₁ h₂ |
| rw [h₁,h₂] at hv |
| refine ⟨padicValRat p y - padicValRat p x, ?_⟩ |
| omega |
| · rw [min_eq_right hl.le] at h₁ h₂ |
| rw [h₁,h₂] at hv |
| exact ⟨padicValRat p y, by omega⟩ |
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| lemma squarefree_support {x r : ℚ} {d D : ℕ} (hd : 0 < d) (hr : r ≠ 0) |
| (hf : Squarefree d) (he : |x| = d*r^2) (hD : D ≠ 0) |
| (hv : ∀ p : ℕ, p.Prime → ¬p ∣ D → Even (padicValRat p x)) : d ∣ D := by |
| apply (Nat.factorization_le_iff_dvd hd.ne' hD).mp |
| intro p |
| by_cases hp : p.Prime |
| · by_cases hpd : p ∣ d |
| · have hdd := Nat.factorization_eq_one_of_squarefree hf hp hpd |
| have hpd' : p ∣ D := by |
| by_contra hpD |
| have : Fact p.Prime := ⟨hp⟩ |
| have hxval := congrArg (padicValRat p) he |
| have habs : padicValRat p |x| = padicValRat p x := by |
| rcases le_or_gt 0 x with hs | hs |
| · rw [abs_of_nonneg hs] |
| · rw [abs_of_neg hs, padicValRat.neg] |
| rw [habs, padicValRat.mul (by exact_mod_cast hd.ne') (pow_ne_zero _ hr), |
| padicValRat.of_nat, padicValRat.pow, ← Nat.factorization_def d hp, hdd] at hxval |
| obtain ⟨k,hk⟩ := hv p hp hpD |
| omega |
| rw [hdd] |
| exact (hp.dvd_iff_one_le_factorization hD).mp hpd' |
| · simp [Nat.factorization_eq_zero_of_not_dvd hpd] |
| · simp [Nat.factorization_eq_zero_of_not_prime d hp] |
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| lemma supported_decomposition {x : ℚ} {D : ℕ} (hx : x ≠ 0) (hD : D ≠ 0) |
| (hv : ∀ p : ℕ, p.Prime → ¬p ∣ D → Even (padicValRat p x)) : |
| ∃ (d : ℕ) (r : ℚ), d ∣ D ∧ r ≠ 0 ∧ (x=d*r^2 ∨ x=-(d*r^2)) := by |
| obtain ⟨d,r,hd,hr,hf,he⟩ := squarefree_decomposition hx |
| refine ⟨d,r,squarefree_support hd hr hf he hD hv,hr,?_⟩ |
| rcases le_or_gt 0 x with hs | hs |
| · exact Or.inl (by simpa [abs_of_nonneg hs] using he) |
| · exact Or.inr (by rw [abs_of_neg hs] at he; linarith) |
|
|
| end SingleFold.SquareSupport |
|
|
| namespace SingleFold.Descent |
| open WeierstrassCurve WeierstrassCurve.Affine WeierstrassCurve.Affine.Point |
| open SquareSupport |
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| lemma supported_zero (P : E.Point) : ∃ (d : ℕ) (r : ℚ), d ∣ 5 ∧ r ≠ 0 ∧ |
| (alphaVal 0 P=d*r^2 ∨ alphaVal 0 P=-(d*r^2)) := by |
| cases P with |
| | zero => exact ⟨1,1,by norm_num,by norm_num,Or.inl (by norm_num [alphaVal])⟩ |
| | some x y h => |
| by_cases hx : x=0 |
| · exact ⟨1,5,by norm_num,by norm_num,Or.inr (by norm_num [alphaVal,adjusted,hx])⟩ |
| by_cases hx5 : x=5 |
| · exact ⟨5,1,by norm_num,by norm_num,Or.inl (by norm_num [alphaVal,adjusted,hx5])⟩ |
| by_cases hxm5 : x= -5 |
| · exact ⟨5,1,by norm_num,by norm_num,Or.inr (by norm_num [alphaVal,adjusted,hxm5])⟩ |
| simp only [alphaVal_some, sub_zero, adjusted, ite_eq_right hx] |
| apply supported_decomposition hx (by norm_num) |
| intro p hp hp5 |
| have : Fact p.Prime := ⟨hp⟩ |
| have hv : padicValRat p (5:ℚ)=0 := by |
| rw [show (5:ℚ)=(5:ℕ) from rfl, padicValRat.of_nat, padicValNat.eq_zero_of_not_dvd hp5] |
| rfl |
| apply even_val_triple (a := -5) (b := 5) hx (by simpa only [sub_eq_add_neg] using sub_ne_zero.mpr hx5) |
| (by intro hh; apply hxm5; linarith) (by norm_num) (by norm_num) (by simpa using hv) hv |
| have he := equation h |
| nlinarith only [he] |
|
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| lemma supported_five (P : E.Point) : ∃ (d : ℕ) (r : ℚ), d ∣ 10 ∧ r ≠ 0 ∧ |
| (alphaVal 5 P=d*r^2 ∨ alphaVal 5 P=-(d*r^2)) := by |
| cases P with |
| | zero => exact ⟨1,1,by norm_num,by norm_num,Or.inl (by norm_num [alphaVal])⟩ |
| | some x y h => |
| by_cases hx : x=0 |
| · exact ⟨5,1,by norm_num,by norm_num,Or.inr (by norm_num [alphaVal,adjusted,hx])⟩ |
| by_cases hx5 : x=5 |
| · exact ⟨2,5,by norm_num,by norm_num,Or.inl (by norm_num [alphaVal,adjusted,hx5])⟩ |
| by_cases hxm5 : x= -5 |
| · exact ⟨10,1,by norm_num,by norm_num,Or.inr (by norm_num [alphaVal,adjusted,hxm5])⟩ |
| have hx5' : x-5 ≠ 0 := sub_ne_zero.mpr hx5 |
| simp only [alphaVal_some, adjusted, ite_eq_right hx5'] |
| apply supported_decomposition hx5' (by norm_num) |
| intro p hp hp10 |
| have : Fact p.Prime := ⟨hp⟩ |
| have hv (k : ℕ) (hk : k ∣ 10) : padicValRat p (k:ℚ)=0 := by |
| rw [padicValRat.of_nat, padicValNat.eq_zero_of_not_dvd (fun hh => hp10 (hh.trans hk))] |
| rfl |
| apply even_val_triple (a := 5) (b := 10) hx5' (by simpa using hx) |
| (by intro hh; apply hxm5; linarith) (by norm_num) (by norm_num) |
| (hv 5 (by norm_num)) (hv 10 (by norm_num)) |
| have he := equation h |
| nlinarith only [he] |
|
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| lemma classOf_mul_sq {d r : ℚ} (hd : d ≠ 0) (hr : r ≠ 0) : |
| classOf (d*r^2)=classOf d := by |
| rw [pow_two, classOf_mul hd (mul_ne_zero hr hr), classOf_square, mul_one] |
|
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| lemma alpha_zero_classes (P : E.Point) : |
| classOf (alphaVal 0 P) = classOf 1 ∨ classOf (alphaVal 0 P) = classOf (-1) ∨ |
| classOf (alphaVal 0 P) = classOf 5 ∨ classOf (alphaVal 0 P) = classOf (-5) := by |
| obtain ⟨d,r,hd,hr,he | he⟩ := supported_zero P |
| all_goals rcases (Nat.dvd_prime (by norm_num : Nat.Prime 5)).mp hd with rfl | rfl |
| · left; rw [he]; exact classOf_mul_sq (by norm_num) hr |
| · right; right; left; rw [he]; exact classOf_mul_sq (by norm_num) hr |
| · right; left; rw [he, ← neg_mul]; exact classOf_mul_sq (by norm_num) hr |
| · right; right; right; rw [he, ← neg_mul]; exact classOf_mul_sq (by norm_num) hr |
|
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| end SingleFold.Descent |
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| end OAI |
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