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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 | import OAI.NumberTheory.SingleFold.QuadraticHeight
namespace OAI
namespace SingleFold.HeightDescent
universe u
variable {A : Type u} [AddCommGroup A]
theorem finitely_generated (h : A → ℝ) [Northcott h]
(hn : ∀ P, 0 ≤ h P)
(hp : ∀ P Q, h (P+Q)+h (P-Q)=2*h P+2*h Q)
(hd : ∀ P, h ((2:ℕ) • P)=4*h P)
(F : Finset A) (hF : ∀ P, ∃ R ∈ F, ∃ Q, P=R+(2:ℕ) • Q) : AddGroup.FG A := by
classical
obtain ⟨B,hB⟩ := (F.finite_toSet.image h).bddAbove
let S : Set A := {P | h P ≤ B}
have hS : S.Finite := Northcott.finite_le B
have hR : ∀ R ∈ F, R ∈ AddSubgroup.closure S := by
intro R hR
apply AddSubgroup.subset_closure
exact hB ⟨R,hR,rfl⟩
have htop : AddSubgroup.closure S = ⊤ := by
apply top_unique
intro P _
by_contra hh
obtain ⟨P,hP,hmin⟩ := Northcott.exists_min_image h {P | P ∉ AddSubgroup.closure S} ⟨P,hh⟩
have hPl : B < h P := by
by_contra hle
exact hP (AddSubgroup.subset_closure (le_of_not_gt hle))
obtain ⟨R,hRF,Q,hPQ⟩ := hF P
have hRB : h R ≤ B := hB ⟨R,hRF,rfl⟩
have he : P-R=(2:ℕ) • Q := by rw [hPQ]; abel
have hdQ : 4*h Q ≤ 2*h P+2*h R := by
have hpara := hp P R
rw [he,hd] at hpara
linarith [hn (P+R)]
have hsmall : h Q < h P := by linarith
have hQ : Q ∈ AddSubgroup.closure S := by
by_contra hQ
have := hmin Q hQ
linarith
apply hP
rw [hPQ]
exact (AddSubgroup.closure S).add_mem (hR R hRF)
((AddSubgroup.closure S).nsmul_mem hQ 2)
exact ⟨hS.toFinset, by simpa only [Set.Finite.coe_toFinset] using htop⟩
end SingleFold.HeightDescent
namespace SingleFold.CurveRank
open Descent
instance : Northcott CurveHeight.canonicalHeight where
finite_le B := by
obtain ⟨c,hc,hbound⟩ := CurveHeight.uniform_height
apply (Northcott.finite_le (h := CurveHeight.naiveHeight) (2*B+c)).subset
intro P hP
have hh := (abs_le.mp (hbound P)).2
change CurveHeight.canonicalHeight P ≤ B at hP
change CurveHeight.naiveHeight P ≤ 2*B+c
linarith
lemma canonical_parallelogram (P Q : E.Point) :
CurveHeight.canonicalHeight (P+Q) + CurveHeight.canonicalHeight (P-Q) =
2*CurveHeight.canonicalHeight P + 2*CurveHeight.canonicalHeight Q := by
obtain ⟨C,hC⟩ := CurveHeight.approximate_parallelogram
have hh := QuadraticLimit.limit_parallelogram CurveHeight.naiveHeight C
CurveHeight.naiveHeight_zero hC P Q
unfold CurveHeight.canonicalHeight
convert congrArg (fun x : ℝ => (1/2:ℝ)*x) hh using 1 <;> ring
noncomputable def representatives : Finset E.Point := by
classical
exact {0,T0,T5,T0+T5} ∪ (({0,T0,T5,T0+T5} : Finset E.Point).image (· + G))
lemma representatives_card : representatives.card ≤ 8 := by
classical
have hu := Finset.card_union_le ({0,T0,T5,T0+T5} : Finset E.Point)
(({0,T0,T5,T0+T5} : Finset E.Point).image (· + G))
have hi := Finset.card_image_le (s := ({0,T0,T5,T0+T5} : Finset E.Point)) (f := (· + G))
have hc : ({0,T0,T5,T0+T5} : Finset E.Point).card ≤ 4 := by
exact (Finset.card_insert_le ..).trans (by
have h1 := Finset.card_insert_le T0 ({T5,T0+T5} : Finset E.Point)
have h2 := Finset.card_insert_le T5 ({T0+T5} : Finset E.Point)
simp only [Finset.card_singleton] at h2
omega)
unfold representatives
omega
lemma representatives_cover (P : E.Point) : ∃ R ∈ representatives, ∃ Q, P=R+(2:ℕ) • Q := by
classical
obtain ⟨S,b,Q,hS,hP⟩ := mod_two_cover P
have hmem : S ∈ ({0,T0,T5,T0+T5} : Finset E.Point) := by simpa using hS
refine ⟨S+(if b then G else 0),?_,Q,hP⟩
cases b
· simpa only [representatives,Bool.false_eq_true,↓reduceIte,add_zero] using Finset.mem_union_left
(({0,T0,T5,T0+T5} : Finset E.Point).image (· + G)) hmem
· exact Finset.mem_union_right _ (Finset.mem_image.mpr ⟨S,hmem,rfl⟩)
instance : AddGroup.FG E.Point :=
HeightDescent.finitely_generated CurveHeight.canonicalHeight CurveHeight.canonicalHeight_nonneg
canonical_parallelogram
(fun P => by convert CurveHeight.canonicalHeight_nsmul P 2 using 1; norm_num)
representatives representatives_cover
end SingleFold.CurveRank
end OAI
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