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namespace OAI
noncomputable section
namespace Problem355.DiagonalDetImage
open Matrix
variable {R : Type*} [CommRing R]
def diagonalThree (D E : R) : Matrix (Fin 3) (Fin 3) R :=
Matrix.diagonal ![1, D, E]
def finalDiagonalUnit (u : Rˣ) : Matrix.GeneralLinearGroup (Fin 3) R :=
Matrix.GeneralLinearGroup.mk'' (Matrix.diagonal ![1, 1, (u : R)]) (by
simp [Matrix.det_diagonal, Fin.prod_univ_succ])
@[simp] theorem finalDiagonalUnit_val (u : Rˣ) :
(finalDiagonalUnit u : Matrix (Fin 3) (Fin 3) R) =
Matrix.diagonal ![1, 1, (u : R)] := by
simp [finalDiagonalUnit]
@[simp] theorem finalDiagonalUnit_det (u : Rˣ) :
Matrix.GeneralLinearGroup.det (finalDiagonalUnit u) = u := by
apply Units.ext
simp [Matrix.GeneralLinearGroup.det, Matrix.det_diagonal, Fin.prod_univ_succ]
theorem finalDiagonalUnit_fixes (D E : R) (u : Rˣ)
(hu : ((u : R) - 1) * E = 0) :
(finalDiagonalUnit u : Matrix (Fin 3) (Fin 3) R) * diagonalThree D E =
diagonalThree D E := by
have huE : (u : R) * E = E := by
simpa [sub_mul, sub_eq_zero] using hu
rw [finalDiagonalUnit_val, diagonalThree, Matrix.diagonal_mul_diagonal]
congr 1
funext i
fin_cases i <;> simp [huE]
def determinantValues (D E : R) : Set Rˣ :=
{u | ∃ g : Matrix.GeneralLinearGroup (Fin 3) R,
(g : Matrix (Fin 3) (Fin 3) R) * diagonalThree D E = diagonalThree D E ∧
Matrix.GeneralLinearGroup.det g = u}
theorem mem_determinantValues (D E : R) (u : Rˣ)
(hu : ((u : R) - 1) * E = 0) : u ∈ determinantValues D E :=
⟨finalDiagonalUnit u, finalDiagonalUnit_fixes D E u hu, finalDiagonalUnit_det u⟩
theorem card_determinantValues_lower [Fintype R] (D E : R) :
Nat.card {u : Rˣ // ((u : R) - 1) * E = 0} ≤
Nat.card (determinantValues D E) := by
classical
apply Nat.card_le_card_of_injective
(fun u => (⟨u.val, mem_determinantValues D E u.val u.property⟩ : determinantValues D E))
intro u v h
apply Subtype.ext
exact congrArg (fun w : determinantValues D E => w.val) h
def annihilatorEquivCongruenceUnits (E : R)
(hunit : ∀ x : R, x * E = 0 → IsUnit (1 + x)) :
{x : R // x * E = 0} ≃ {u : Rˣ // ((u : R) - 1) * E = 0} where
toFun x := ⟨(hunit x.val x.property).unit, by
rw [IsUnit.unit_spec]
simpa using x.property⟩
invFun u := ⟨(u.val : R) - 1, u.property⟩
left_inv x := by
apply Subtype.ext
simp only [IsUnit.unit_spec, add_sub_cancel_left]
right_inv u := by
apply Subtype.ext
apply Units.ext
simp only [IsUnit.unit_spec]
ring
theorem isUnit_one_add_of_annihilates_prime_power
{p k e : ℕ} (hp : p.Prime) (he : e < k)
(x : ZMod (p ^ k)) (hx : x * (p ^ e : ℕ) = 0) :
IsUnit (1 + x) := by
have hp0 : 0 < p := hp.pos
have hk : 0 < k := lt_of_le_of_lt (Nat.zero_le e) he
let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩
have hcast : ((x.val * p ^ e : ℕ) : ZMod (p ^ k)) = 0 := by
simpa only [Nat.cast_mul, ZMod.natCast_zmod_val] using hx
have hd : p ^ k ∣ x.val * p ^ e :=
(ZMod.natCast_eq_zero_iff _ _).mp hcast
have hd' : p ^ (k - e) ∣ x.val := by
apply Nat.dvd_of_mul_dvd_mul_right (pow_pos hp0 e)
simpa only [← pow_add, Nat.sub_add_cancel he.le] using hd
have hpval : p ∣ x.val :=
(dvd_pow_self p (by omega : k - e ≠ 0)).trans hd'
have hpnot : ¬ p ∣ 1 + x.val := by
intro h
have hpone : p ∣ 1 := (Nat.dvd_add_iff_left hpval).mpr h
exact hp.not_dvd_one hpone
have hu : IsUnit ((1 + x.val : ℕ) : ZMod (p ^ k)) :=
(ZMod.isUnit_natCast_iff_not_dvd_pow hp hk).mpr hpnot
simpa only [Nat.cast_add, Nat.cast_one, ZMod.natCast_zmod_val] using hu
theorem card_congruenceUnits_eq_annihilator
{p k e : ℕ} (hp : p.Prime) (he : e < k) :
Nat.card {u : (ZMod (p ^ k))ˣ //
((u : ZMod (p ^ k)) - 1) * (p ^ e : ℕ) = 0} =
Nat.card {x : ZMod (p ^ k) // x * (p ^ e : ℕ) = 0} := by
exact (Nat.card_congr (annihilatorEquivCongruenceUnits ((p ^ e : ℕ) : ZMod (p ^ k))
(isUnit_one_add_of_annihilates_prime_power hp he))).symm
theorem card_congruenceUnits_prime_power
{p k e : ℕ} (hp : p.Prime) (he : e < k) :
Nat.card {u : (ZMod (p ^ k))ˣ //
((u : ZMod (p ^ k)) - 1) * (p ^ e : ℕ) = 0} = p ^ e := by
let : NeZero p := ⟨hp.ne_zero⟩
rw [card_congruenceUnits_eq_annihilator hp he]
simpa only [Nat.cast_pow, mul_comm] using
Problem355.zmod_card_pow_mul_eq_zero p k e he.le
theorem determinantValues_zero (D : R) :
determinantValues D 0 = Set.univ := by
apply Set.eq_univ_of_forall
intro u
exact mem_determinantValues D 0 u (by simp)
theorem card_determinantValues_prime_power_zero
{p k : ℕ} (hp : p.Prime) (hk : 0 < k) (D : ZMod (p ^ k)) :
Nat.card (determinantValues D ((p ^ k : ℕ) : ZMod (p ^ k))) =
p ^ (k - 1) * (p - 1) := by
let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩
simp only [ZMod.natCast_self, determinantValues_zero]
rw [Nat.card_congr (Equiv.Set.univ _)]
rw [Nat.card_eq_fintype_card, ZMod.card_units_eq_totient,
Nat.totient_prime_pow hp hk]
theorem card_determinantValues_prime_power_lower
{p k e : ℕ} (hp : p.Prime) (he : e < k) (D : ZMod (p ^ k)) :
p ^ e ≤ Nat.card (determinantValues D ((p ^ e : ℕ) : ZMod (p ^ k))) := by
let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩
simpa only [card_congruenceUnits_prime_power hp he] using
card_determinantValues_lower D ((p ^ e : ℕ) : ZMod (p ^ k))
theorem card_determinantValues_ratio
{p k e : ℕ} (hp : p.Prime) (he : e ≤ k) (D : ZMod (p ^ k)) :
p ^ e * Nat.card (ZMod (p ^ k))ˣ ≤
p ^ k * Nat.card (determinantValues D ((p ^ e : ℕ) : ZMod (p ^ k))) := by
let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩
rcases he.eq_or_lt with rfl | he
· simp only [ZMod.natCast_self, determinantValues_zero]
rw [Nat.card_congr (Equiv.Set.univ _)]
· have h1 := card_determinantValues_prime_power_lower hp he D
have h2 : Nat.card (ZMod (p ^ k))ˣ ≤ p ^ k := by
simpa only [Nat.card_zmod] using
Nat.card_le_card_of_injective
(fun u : (ZMod (p ^ k))ˣ => (u : ZMod (p ^ k))) Units.val_injective
exact (Nat.mul_le_mul_left (p ^ e) h2).trans
(by simpa only [Nat.mul_comm] using Nat.mul_le_mul_left (p ^ k) h1)
theorem determinantValues_eq_stabilizer_map (D E : R) :
determinantValues D E =
((MulAction.stabilizer (Matrix.GeneralLinearGroup (Fin 3) R)
(diagonalThree D E)).map Matrix.GeneralLinearGroup.det : Set Rˣ) := by
ext u
simp only [determinantValues, Set.mem_ofPred_eq, SetLike.mem_coe,
Subgroup.mem_map, MulAction.mem_stabilizer_iff]
rfl
theorem determinant_mem_of_diagonal_form
(D E : R) (P Q : Matrix.GeneralLinearGroup (Fin 3) R)
(C : Matrix (Fin 3) (Fin 3) R)
(hC : C = (P : Matrix (Fin 3) (Fin 3) R) * diagonalThree D E *
(Q : Matrix (Fin 3) (Fin 3) R))
(u : Rˣ) (hu : u ∈ determinantValues D E) :
u ∈ (MulAction.stabilizer (Matrix.GeneralLinearGroup (Fin 3) R) C).map
Matrix.GeneralLinearGroup.det := by
rcases hu with ⟨g, hg, hdet⟩
apply Subgroup.mem_map.mpr
refine ⟨P * g * P⁻¹, ?_, ?_⟩
· apply MulAction.mem_stabilizer_iff.mpr
change (↑(P * g * P⁻¹) : Matrix (Fin 3) (Fin 3) R) * C = C
rw [hC]
simp only [Units.val_mul]
calc
((P : Matrix (Fin 3) (Fin 3) R) * ↑g * ↑(P⁻¹)) *
(↑P * diagonalThree D E * ↑Q) =
(P : Matrix (Fin 3) (Fin 3) R) *
(↑g * diagonalThree D E) * ↑Q := by
have hc : (↑(P⁻¹) : Matrix (Fin 3) (Fin 3) R) *
((↑P : Matrix (Fin 3) (Fin 3) R) * (diagonalThree D E * (Q : Matrix (Fin 3) (Fin 3) R))) =
diagonalThree D E * (Q : Matrix (Fin 3) (Fin 3) R) := by
rw [← Matrix.mul_assoc, P.inv_mul, Matrix.one_mul]
simpa only [Matrix.mul_assoc] using
congrArg (fun M : Matrix (Fin 3) (Fin 3) R =>
(P : Matrix (Fin 3) (Fin 3) R) * ((g : Matrix (Fin 3) (Fin 3) R) * M)) hc
_ = _ := by rw [hg]
· simp only [map_mul, map_inv]
rw [mul_right_comm, mul_inv_cancel, one_mul, hdet]
theorem card_stabilizer_det_image_ratio_of_diagonal_form
{p k e : ℕ} (hp : p.Prime) (he : e ≤ k)
(D : ZMod (p ^ k))
(P Q : Matrix.GeneralLinearGroup (Fin 3) (ZMod (p ^ k)))
(C : Matrix (Fin 3) (Fin 3) (ZMod (p ^ k)))
(hC : C = (P : Matrix (Fin 3) (Fin 3) (ZMod (p ^ k))) *
diagonalThree D ((p ^ e : ℕ) : ZMod (p ^ k)) *
(Q : Matrix (Fin 3) (Fin 3) (ZMod (p ^ k)))) :
p ^ e * Nat.card (ZMod (p ^ k))ˣ ≤
p ^ k * Nat.card
((MulAction.stabilizer
(Matrix.GeneralLinearGroup (Fin 3) (ZMod (p ^ k))) C).map
Matrix.GeneralLinearGroup.det) := by
let : NeZero (p ^ k) := ⟨pow_ne_zero _ hp.ne_zero⟩
apply (card_determinantValues_ratio hp he D).trans
apply Nat.mul_le_mul_left
apply Nat.card_le_card_of_injective
(fun u : determinantValues D ((p ^ e : ℕ) : ZMod (p ^ k)) =>
(⟨u.val, determinant_mem_of_diagonal_form D _ P Q C hC u.val u.property⟩ :
(MulAction.stabilizer
(Matrix.GeneralLinearGroup (Fin 3) (ZMod (p ^ k))) C).map
Matrix.GeneralLinearGroup.det))
intro u v h
apply Subtype.ext
exact congrArg (fun w :
((MulAction.stabilizer
(Matrix.GeneralLinearGroup (Fin 3) (ZMod (p ^ k))) C).map
Matrix.GeneralLinearGroup.det) => w.val) h
end Problem355.DiagonalDetImage
end
end OAI
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