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5bdeca1 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.NumberTheory.SingleFold.DescentMap
namespace OAI
namespace SingleFold.SquareSupport
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']
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⟩
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]
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
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]
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]
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]
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
end SingleFold.Descent
end OAI
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