| import OAI.NumberTheory.SingleFold.SquareSupport |
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| namespace OAI |
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| namespace SingleFold.LocalFive |
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| lemma mod_five_anisotropic (c : ℤ) (hc : c=2 ∨ c= -2) (a b : ℤ) (h : (5:ℤ) ∣ a^2-c*b^2) : |
| (5:ℤ) ∣ a ∧ (5:ℤ) ∣ b := by |
| have hh : (a:ZMod 5)^2-(c:ZMod 5)*(b:ZMod 5)^2=0 := by |
| exact_mod_cast (ZMod.intCast_zmod_eq_zero_iff_dvd (a^2-c*b^2) 5).mpr h |
| have hz : (a:ZMod 5)=0 ∧ (b:ZMod 5)=0 := by |
| rcases hc with rfl | rfl |
| · exact (by decide : ∀ a b : ZMod 5, a^2-2*b^2=0 → a=0 ∧ b=0) _ _ (by simpa using hh) |
| · exact (by decide : ∀ a b : ZMod 5, a^2-(-2)*b^2=0 → a=0 ∧ b=0) _ _ (by simpa using hh) |
| exact ⟨(ZMod.intCast_zmod_eq_zero_iff_dvd a 5).mp hz.1, |
| (ZMod.intCast_zmod_eq_zero_iff_dvd b 5).mp hz.2⟩ |
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| lemma five_dvd_of_sq {a : ℤ} (h : (5:ℤ) ∣ a^2) : (5:ℤ) ∣ a := |
| (show Prime (5:ℤ) by norm_num).dvd_of_dvd_pow h |
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| lemma no_norm_five_int (c : ℤ) (hc : c=2 ∨ c= -2) (N : ℕ) : ∀ R S : ℤ, R^2-c*S^2=5*(N:ℤ)^2 → N=0 := by |
| induction N using Nat.strong_induction_on with |
| | h N ih => |
| intro R S he |
| by_cases hN : N=0 |
| · exact hN |
| obtain ⟨hr,hs⟩ := mod_five_anisotropic c hc R S ⟨(N:ℤ)^2, he⟩ |
| obtain ⟨r,hr⟩ := hr |
| obtain ⟨s,hs⟩ := hs |
| have hd : (5:ℤ) ∣ (N:ℤ)^2 := ⟨r^2-c*s^2, by rw [hr,hs] at he; nlinarith only [he]⟩ |
| have hd' : 5 ∣ N := by exact_mod_cast five_dvd_of_sq hd |
| obtain ⟨n,hn⟩ := hd' |
| have hnl : n < N := by omega |
| have he' : r^2-c*s^2=5*(n:ℤ)^2 := by |
| rw [hr,hs,hn] at he |
| push_cast at he |
| nlinarith only [he] |
| have hn0 := ih n hnl r s he' |
| omega |
|
|
| lemma no_norm_five (c : ℤ) (hc : c=2 ∨ c= -2) (r s : ℚ) : r^2-(c:ℚ)*s^2 ≠ 5 := by |
| intro h |
| let N := r.den*s.den |
| let R : ℤ := r.num*s.den |
| let S : ℤ := s.num*r.den |
| have he : R^2-c*S^2=5*(N:ℤ)^2 := by |
| have h' : (R:ℚ)^2-(c:ℚ)*(S:ℚ)^2=5*(N:ℚ)^2 := by |
| dsimp [R,S,N] |
| push_cast |
| have hr := r.num_div_den |
| have hs := s.num_div_den |
| rw [←hr,←hs] at h |
| field_simp at h |
| nlinarith only [h] |
| exact_mod_cast h' |
| have hz := no_norm_five_int c hc N R S he |
| exact (Nat.mul_ne_zero r.den_ne_zero s.den_ne_zero) hz |
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|
| lemma no_norm_ten (r s : ℚ) : r^2-10*s^2 ≠ 5 := by |
| intro h |
| have hr : r ≠ 0 := by intro hh; rw [hh] at h; nlinarith [sq_nonneg s] |
| have h' : (5/r)^2-(-2:ℤ)*(5*s/r)^2=5 := by |
| push_cast |
| field_simp |
| nlinarith only [h] |
| exact no_norm_five (-2) (Or.inr rfl) (5/r) (5*s/r) h' |
|
|
| end SingleFold.LocalFive |
|
|
| namespace SingleFold.Descent |
| open WeierstrassCurve WeierstrassCurve.Affine WeierstrassCurve.Affine.Point |
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| lemma alpha_five_when_zero (P : E.Point) (h0 : classOf (alphaVal 0 P)=1) : |
| classOf (alphaVal 5 P)=1 ∨ classOf (alphaVal 5 P)=classOf 5 := by |
| cases P with |
| | zero => exact Or.inl (by simp [←zero_def]) |
| | some x y h => |
| have hsq := (classOf_eq_one_iff (alphaVal_ne_zero 0 (by norm_num) (some x y h))).mp h0 |
| by_cases hx : x=0 |
| · have hs : IsSquare (-25:ℚ) := by |
| simp only [alphaVal,adjusted,hx,sub_self] at hsq |
| convert hsq using 1; norm_num |
| obtain ⟨r,hr⟩ := hs |
| nlinarith [sq_nonneg r] |
| have hs : IsSquare x := by simpa [alphaVal,adjusted,hx] using hsq |
| obtain ⟨r,hr⟩ := hs |
| have hxp : 0 < x := lt_of_le_of_ne (by nlinarith [sq_nonneg r]) (Ne.symm hx) |
| have hx5 : x ≠ 5 := by |
| intro he |
| exact not_isSquare_prime 5 (by norm_num) (by change IsSquare (5:ℚ); rw [←he]; exact ⟨r,hr⟩) |
| have hx5p : 5 < x := by |
| have he := equation h |
| by_contra hn |
| have hl : x < 5 := lt_of_le_of_ne (le_of_not_gt hn) hx5 |
| have hm : x*(5-x)*(x+5)>0 := mul_pos (mul_pos hxp (sub_pos.mpr hl)) (by linarith) |
| nlinarith [sq_nonneg y] |
| obtain ⟨d,s,hd,hs0,he⟩ := supported_five (some x y h) |
| have ha : alphaVal 5 (some x y h)=x-5 := by simp [alphaVal,adjusted,sub_ne_zero.mpr hx5] |
| rw [ha] at he ⊢ |
| have hdpos : 0 < d := Nat.pos_of_ne_zero (by intro hh; subst d; norm_num at hd) |
| have hnneg : ¬ x-5= -((d:ℚ)*s^2) := by |
| have hdq : 0 ≤ (d:ℚ) := Nat.cast_nonneg d |
| have hnon := mul_nonneg hdq (sq_nonneg s) |
| intro hh |
| linarith |
| have he := he.resolve_right hnneg |
| have hdle := Nat.le_of_dvd (by norm_num : 0 < 10) hd |
| interval_cases d <;> norm_num at hd |
| · left |
| rw [he] |
| simpa using classOf_mul_sq (d:=1) (by norm_num) hs0 |
| · exfalso |
| apply LocalFive.no_norm_five 2 (Or.inl rfl) r s |
| norm_num at he ⊢ |
| nlinarith only [hr,he] |
| · right |
| rw [he] |
| exact classOf_mul_sq (by norm_num) hs0 |
| · exfalso |
| apply LocalFive.no_norm_ten r s |
| norm_num at he |
| nlinarith only [hr,he] |
|
|
| noncomputable def a0 : E.Point →+ Additive SquareClass := alpha 0 (by norm_num) (by norm_num) |
| noncomputable def a5 : E.Point →+ Additive SquareClass := alpha 5 (by norm_num) (by norm_num) |
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| def T0 : E.Point := some 0 0 (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E])) |
| def T5 : E.Point := some 5 0 (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E])) |
| def G : E.Point := some (25/4) (75/8) (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E])) |
|
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| lemma a0_T0 : a0 T0 = Additive.ofMul (classOf (-1)) := by |
| change classOf _ = _ |
| have hh := classOf_mul_sq (d := -1) (r := 5) (by norm_num) (by norm_num) |
| norm_num [T0,alphaVal,adjusted] at * |
| exact hh |
| lemma a0_T5 : a0 T5 = Additive.ofMul (classOf 5) := by |
| change classOf (alphaVal 0 T5) = classOf 5 |
| norm_num [T5,alphaVal,adjusted] |
| lemma a0_G : a0 G = 0 := by |
| change classOf _ = 1 |
| have hh := classOf_mul_sq (d := 1) (r := 5/2) (by norm_num) (by norm_num) |
| norm_num [G,alphaVal,adjusted] at * |
| exact hh |
| lemma a5_G : a5 G = Additive.ofMul (classOf 5) := by |
| change classOf _ = _ |
| have hh := classOf_mul_sq (d := 5) (r := 1/2) (by norm_num) (by norm_num) |
| norm_num [G,alphaVal,adjusted] at * |
| exact hh |
|
|
| lemma normalize_a0 (P : E.Point) : ∃ S : E.Point, |
| (S=0 ∨ S=T0 ∨ S=T5 ∨ S=T0+T5) ∧ a0 (P-S)=0 := by |
| obtain h | h | h | h := alpha_zero_classes P |
| · refine ⟨0,Or.inl rfl,?_⟩ |
| change classOf (alphaVal 0 (P-0))=1 |
| simpa only [sub_zero,classOf_one] using h |
| · refine ⟨T0,Or.inr (Or.inl rfl),?_⟩ |
| rw [map_sub,a0_T0] |
| exact sub_eq_zero.mpr h |
| · refine ⟨T5,Or.inr (Or.inr (Or.inl rfl)),?_⟩ |
| rw [map_sub,a0_T5] |
| exact sub_eq_zero.mpr h |
| · refine ⟨T0+T5,Or.inr (Or.inr (Or.inr rfl)),?_⟩ |
| rw [map_sub,map_add,a0_T0,a0_T5] |
| apply sub_eq_zero.mpr |
| change classOf (alphaVal 0 P)=classOf (-1)*classOf 5 |
| rw [←classOf_mul (by norm_num) (by norm_num)] |
| norm_num |
| exact h |
|
|
| theorem mod_two_cover (P : E.Point) : ∃ S : E.Point, ∃ b : Bool, ∃ Q : E.Point, |
| (S=0 ∨ S=T0 ∨ S=T5 ∨ S=T0+T5) ∧ P = (S + if b then G else 0) + (2:ℕ) • Q := by |
| obtain ⟨S,hS,h0⟩ := normalize_a0 P |
| have h5 := alpha_five_when_zero (P-S) h0 |
| rcases h5 with h5 | h5 |
| · have hk : kummer (P-S)=0 := by |
| apply Prod.ext |
| · exact h0 |
| · exact h5 |
| obtain ⟨Q,hQ⟩ := (kummer_eq_zero_iff _).mp hk |
| refine ⟨S,false,Q,hS,?_⟩ |
| simp only [Bool.false_eq_true, ↓reduceIte, add_zero] |
| rw [hQ] |
| abel |
| · have hk : kummer (P-S-G)=0 := by |
| apply Prod.ext |
| · change a0 (P-S-G)=0 |
| rw [map_sub,h0,a0_G,sub_self] |
| · change a5 (P-S-G)=0 |
| rw [map_sub,a5_G] |
| exact sub_eq_zero.mpr h5 |
| obtain ⟨Q,hQ⟩ := (kummer_eq_zero_iff _).mp hk |
| refine ⟨S,true,Q,hS,?_⟩ |
| simp only [↓reduceIte] |
| rw [hQ] |
| abel |
|
|
| end SingleFold.Descent |
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| end OAI |
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