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67aee1d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Analysis.GroupDeterminants.Defects
namespace OAI
universe uG uA um ur
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
section ModularKernel
variable (k : ℕ) (hk : 0 < k)
def reduceTwo : ZMod (2 ^ k) →+* ZMod 2 :=
ZMod.castHom (dvd_pow_self 2 (Nat.ne_of_gt hk)) (ZMod 2)
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
variable {G : Type uG} [Group G]
def reduceGroupTwo : MonoidAlgebra (ZMod (2 ^ k)) G →+* MonoidAlgebra (ZMod 2) G :=
MonoidAlgebra.mapRingHom G (reduceTwo k hk)
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))
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
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]
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
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)))
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]
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
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
end GroupRingDeterminant
end
end OAI
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