| import Mathlib |
| import OAI.NumberTheory.CubicGram.Coordinates |
|
|
| namespace OAI |
|
|
| /-! The integral basis, multiplication norm and finite residue quotients. -/ |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open Module |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
| def coordinatesMap : (Fin 2 → ℤ) →ₗ[ℤ] Eisenstein where |
| toFun v := ofCoords (v 0) (v 1) |
| map_add' v w := by simp [ofCoords, add_mul]; ring |
| map_smul' n v := by simp [ofCoords]; ring |
|
|
| lemma coordinatesMap_bijective : Function.Bijective coordinatesMap := by |
| constructor |
| · intro v w h |
| have hh := coordinates_unique (congrArg (fun z : Eisenstein => (z : ℂ)) h) |
| funext i |
| fin_cases i |
| · exact hh.1 |
| · exact hh.2 |
| · intro z |
| obtain ⟨a,b,h⟩ := exists_coordinates z |
| exact ⟨![a,b], Subtype.ext h.symm⟩ |
|
|
| def coordinatesEquiv : (Fin 2 → ℤ) ≃ₗ[ℤ] Eisenstein := |
| LinearEquiv.ofBijective coordinatesMap coordinatesMap_bijective |
|
|
| def integerBasis : Basis (Fin 2) ℤ Eisenstein := |
| (Pi.basisFun ℤ (Fin 2)).map coordinatesEquiv |
|
|
| instance : Module.Free ℤ Eisenstein := Module.Free.of_basis integerBasis |
| instance : Module.Finite ℤ Eisenstein := Module.Finite.of_basis integerBasis |
|
|
| @[simp] lemma integerBasis_zero : integerBasis 0 = 1 := by |
| simp [integerBasis, coordinatesEquiv, coordinatesMap, ofCoords, Pi.basisFun_apply] |
|
|
| @[simp] lemma integerBasis_one : integerBasis 1 = omegaE := by |
| simp [integerBasis, coordinatesEquiv, coordinatesMap, ofCoords, Pi.basisFun_apply] |
|
|
| lemma integerBasis_repr_ofCoords (a b : ℤ) : |
| integerBasis.repr (ofCoords a b) = Finsupp.equivFunOnFinite.symm ![a,b] := by |
| apply Finsupp.equivFunOnFinite.injective |
| change coordinatesEquiv.symm (ofCoords a b) = ![a,b] |
| apply coordinatesEquiv.injective |
| simp only [LinearEquiv.apply_symm_apply] |
| rfl |
|
|
| lemma ofCoords_mul (a b c d : ℤ) : |
| ofCoords a b * ofCoords c d = ofCoords (a*c-b*d) (a*d+b*c-b*d) := by |
| apply Subtype.ext |
| change ((a : ℂ) + b * omega) * (c + d * omega) = _ |
| rw [ofCoords_coe] |
| have h : omega^2 = -omega - 1 := by linear_combination omega_quadratic |
| push_cast |
| ring_nf |
| rw [h] |
| ring |
|
|
| lemma ofCoords_mul_omega (a b : ℤ) : ofCoords a b * omegaE = ofCoords (-b) (a-b) := by |
| rw [show omegaE = ofCoords 0 1 by simp [ofCoords], ofCoords_mul] |
| simp |
|
|
| lemma leftMulMatrix_ofCoords (a b : ℤ) : |
| Algebra.leftMulMatrix integerBasis (ofCoords a b) = !![a, -b; b, a-b] := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> |
| simp [Algebra.leftMulMatrix_eq_repr_mul, ofCoords_mul_omega, integerBasis_repr_ofCoords] |
|
|
| lemma algebra_norm_ofCoords (a b : ℤ) : |
| Algebra.norm ℤ (ofCoords a b) = a^2 - a*b + b^2 := by |
| rw [Algebra.norm_eq_matrix_det integerBasis, leftMulMatrix_ofCoords, Matrix.det_fin_two] |
| simp |
| ring |
|
|
| lemma algebra_norm_cast (z : Eisenstein) : (Algebra.norm ℤ z : ℝ) = norm z := by |
| obtain ⟨⟨a,b⟩,rfl⟩ := ofCoords_surjective z |
| rw [algebra_norm_ofCoords, norm, ofCoords_coe, coordinates_norm] |
|
|
| lemma residues_card {z : Eisenstein} (hz : z ≠ 0) : Nat.card (Residues z) = normNat z := by |
| have hdet := Submodule.natAbs_det_basis_change integerBasis |
| ((modulus z).restrictScalars ℤ) (Ideal.basisSpanSingleton integerBasis hz) |
| have hh : integerBasis.det (fun i => |
| ((Ideal.basisSpanSingleton integerBasis hz i : modulus z) : Eisenstein)) = |
| Algebra.norm ℤ z := by |
| rw [Algebra.norm_eq_matrix_det integerBasis, Basis.det_apply] |
| congr 1 |
| ext i j |
| simp [Basis.toMatrix_apply, Ideal.basisSpanSingleton_apply, |
| Algebra.leftMulMatrix_eq_repr_mul] |
| change (integerBasis.det (fun i => |
| ((Ideal.basisSpanSingleton integerBasis hz i : modulus z) : Eisenstein))).natAbs = |
| Nat.card (Residues z) at hdet |
| rw [hh] at hdet |
| have hn : 0 ≤ Algebra.norm ℤ z := by |
| exact_mod_cast (algebra_norm_cast z).symm ▸ norm_nonneg z |
| rw [← hdet] |
| apply Nat.cast_injective (R := ℝ) |
| rw [normNat_cast, ← Int.cast_natCast, Int.natCast_natAbs, abs_of_nonneg hn, algebra_norm_cast] |
|
|
| lemma norm_eq_zero_iff {z : Eisenstein} : norm z = 0 ↔ z = 0 := by |
| rw [norm, Complex.normSq_eq_zero] |
| constructor |
| · exact fun h => Subtype.ext h |
| · rintro rfl |
| rfl |
|
|
| lemma normNat_ne_zero {z : Eisenstein} (hz : z ≠ 0) : normNat z ≠ 0 := by |
| intro h |
| apply hz |
| apply norm_eq_zero_iff.mp |
| rw [← normNat_cast, h, Nat.cast_zero] |
|
|
| lemma finite_residues {z : Eisenstein} (hz : z ≠ 0) : Finite (Residues z) := |
| Nat.finite_of_card_ne_zero (by rw [residues_card hz]; exact normNat_ne_zero hz) |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| end OAI |
|
|
