| import OAI.NumberTheory.SingleFold.QuadraticHeight |
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|
| namespace OAI |
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| namespace SingleFold.HeightDescent |
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|
| universe u |
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| variable {A : Type u} [AddCommGroup A] |
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| 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)) |
|
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| 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 |
| |
