| import OAI.Analysis.GroupDeterminants.Defects |
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| namespace OAI |
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| universe uG uA um ur |
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| noncomputable section |
| open scoped BigOperators |
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| namespace GroupRingDeterminant |
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|
| section ModularKernel |
| variable (k : ℕ) (hk : 0 < k) |
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| def reduceTwo : ZMod (2 ^ k) →+* ZMod 2 := |
| ZMod.castHom (dvd_pow_self 2 (Nat.ne_of_gt hk)) (ZMod 2) |
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| theorem kernel_reduceTwo (a : ZMod (2 ^ k)) (ha : reduceTwo k hk a = 0) : |
| a = 2 * (a.val / 2 : ℕ) := by |
| have : NeZero (2 ^ k) := ⟨pow_ne_zero _ (by decide)⟩ |
| have hv : (a.val : ZMod 2) = 0 := by |
| rw [← map_natCast (reduceTwo k hk), ZMod.natCast_zmod_val] |
| exact ha |
| have hd : 2 ∣ a.val := (ZMod.natCast_eq_zero_iff _ _).mp hv |
| calc |
| a = (a.val : ZMod (2 ^ k)) := (ZMod.natCast_zmod_val a).symm |
| _ = ((2 * (a.val / 2) : ℕ) : ZMod (2 ^ k)) := by rw [Nat.mul_div_cancel' hd] |
| _ = _ := by push_cast; rfl |
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| variable {G : Type uG} [Group G] |
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| def reduceGroupTwo : MonoidAlgebra (ZMod (2 ^ k)) G →+* MonoidAlgebra (ZMod 2) G := |
| MonoidAlgebra.mapRingHom G (reduceTwo k hk) |
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| def halfGroup (a : MonoidAlgebra (ZMod (2 ^ k)) G) : |
| MonoidAlgebra (ZMod (2 ^ k)) G := |
| MonoidAlgebra.ofCoeff (a.coeff.mapRange |
| (fun c => (c.val / 2 : ℕ)) (by simp)) |
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| theorem kernel_reduceGroupTwo (a : MonoidAlgebra (ZMod (2 ^ k)) G) |
| (ha : reduceGroupTwo k hk a = 0) : a = 2 * halfGroup k a := by |
| apply MonoidAlgebra.ext |
| ext g |
| have hg := congrArg (fun c : MonoidAlgebra (ZMod 2) G => c.coeff g) ha |
| have hc : reduceTwo k hk (a.coeff g) = 0 := by |
| simpa only [reduceGroupTwo, MonoidAlgebra.coeff_mapRingHom, |
| MonoidAlgebra.coeff_zero, Finsupp.zero_apply] using hg |
| simpa [two_mul, halfGroup, MonoidAlgebra.coeff_add] using kernel_reduceTwo k hk _ hc |
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| theorem twice_pow_eq_zero {A : Type uA} [Ring A] [Algebra (ZMod (2 ^ k)) A] (v : A) : |
| (2 * v) ^ k = 0 := by |
| have htwo : (2 : ZMod (2 ^ k)) ^ k = 0 := by |
| simpa only [Nat.cast_pow, Nat.cast_ofNat] using ZMod.natCast_self (2 ^ k) |
| have ht : (2 : A) ^ k = 0 := by |
| simpa only [map_pow, map_ofNat, map_zero] using |
| congrArg (algebraMap (ZMod (2 ^ k)) A) htwo |
| have hc : Commute (2 : A) v := by |
| simpa only [Nat.cast_ofNat] using Nat.cast_commute 2 v |
| rw [hc.mul_pow, ht, zero_mul] |
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| theorem matrix_kernel_nilpotent {m : Type um} [Fintype m] [DecidableEq m] |
| (W : Matrix m m (MonoidAlgebra (ZMod (2 ^ k)) G)) |
| (hW : W.map (reduceGroupTwo k hk) = 0) : W ^ k = 0 := by |
| let V : Matrix m m (MonoidAlgebra (ZMod (2 ^ k)) G) := fun i j => halfGroup k (W i j) |
| have hv : W = 2 * V := by |
| rw [two_mul] |
| apply Matrix.ext |
| intro i j |
| have hc : reduceGroupTwo k hk (W i j) = 0 := congrFun (congrFun hW i) j |
| change W i j = halfGroup k (W i j) + halfGroup k (W i j) |
| simpa only [two_mul] using kernel_reduceGroupTwo k hk _ hc |
| rw [hv] |
| exact twice_pow_eq_zero k V |
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| theorem reduceGroupTwo_surjective : |
| Function.Surjective (reduceGroupTwo k hk (G := G)) := |
| MonoidAlgebra.map_surjective _ (ZMod.castHom_surjective (dvd_pow_self 2 (Nat.ne_of_gt hk))) |
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| theorem integerReduction_surjective (q : ℕ) : Function.Surjective |
| (MonoidAlgebra.mapRingHom G (Int.castRingHom (ZMod q))) := |
| MonoidAlgebra.map_surjective _ ZMod.intCast_surjective |
|
|
| end ModularKernel |
|
|
| section ModularFactorization |
| variable {G : Type uG} {m : Type um} {r : Type ur} [Group G] [Fintype m] [DecidableEq m] [Fintype r] |
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| theorem lift_binary_factorization (k : ℕ) (hk : 0 < k) |
| (e : Matrix m m (MonoidAlgebra (ZMod (2 ^ k)) G)) (he : e * e = e) |
| (u : Matrix m r (MonoidAlgebra (ZMod 2) G)) |
| (v : Matrix r m (MonoidAlgebra (ZMod 2) G)) |
| (huv : u * v = e.map (reduceGroupTwo k hk)) : |
| ∃ u' : Matrix m r (MonoidAlgebra (ZMod (2 ^ k)) G), |
| ∃ v' : Matrix r m (MonoidAlgebra (ZMod (2 ^ k)) G), u' * v' = e := by |
| classical |
| have hs := reduceGroupTwo_surjective (G := G) k hk |
| choose uRaw hu₀ using fun i j => hs (u i j) |
| choose vRaw hv₀ using fun i j => hs (v i j) |
| let u₀ : Matrix m r (MonoidAlgebra (ZMod (2 ^ k)) G) := uRaw |
| let v₀ : Matrix r m (MonoidAlgebra (ZMod (2 ^ k)) G) := vRaw |
| have hu : Matrix.map u₀ (reduceGroupTwo k hk) = u := Matrix.ext hu₀ |
| have hv : Matrix.map v₀ (reduceGroupTwo k hk) = v := Matrix.ext hv₀ |
| apply lift_rectangular_factorization e he u₀ v₀ k |
| apply matrix_kernel_nilpotent k hk |
| rw [Matrix.map_sub _ (map_sub _) _ _] |
| simp only [Matrix.map_mul, hu, hv] |
| rw [Matrix.mul_assoc (e.map _) u, huv] |
| have he' : e.map (reduceGroupTwo k hk) * e.map (reduceGroupTwo k hk) = |
| e.map (reduceGroupTwo k hk) := by rw [← Matrix.map_mul, he] |
| rw [he', he', sub_self] |
|
|
| end ModularFactorization |
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| theorem exists_binary_modulus (D : ℕ) (hD : 0 < D) (hodd : Odd D) : |
| ∃ k : ℕ, 0 < k ∧ 1 < 2 ^ k ∧ Nat.ModEq D (2 ^ k) 1 := by |
| have hk : 0 < D.totient := Nat.totient_pos.mpr hD |
| refine ⟨D.totient, hk, one_lt_pow₀ (by decide) (Nat.ne_of_gt hk), ?_⟩ |
| exact Nat.ModEq.pow_totient (Nat.coprime_two_right.mpr hodd).symm |
|
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| end GroupRingDeterminant |
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| end |
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| end OAI |
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