| import OAI.LinearAlgebra.CirculantHadamard.LocalComparisonInduction |
| import OAI.LinearAlgebra.CirculantHadamard.CyclotomicFactorDivision |
| import OAI.LinearAlgebra.CirculantHadamard.CyclicEvaluationProjection |
| import OAI.LinearAlgebra.CirculantHadamard.RamifiedLocal |
|
|
| namespace OAI |
|
|
| universe uA |
|
|
| /-! |
| # Local character comparison in the actual cyclotomic quotient |
|
|
| Starting with a characteristic-zero DVR whose maximal ideal is generated by |
| the rational prime `p`, all ramified rings needed by the induction are |
| constructed as the actual cyclotomic quotients. No projection-divisibility, |
| comparison, or existence-of-valuation premise is taken from the caller. |
|
|
| The conclusion uses membership in the ideal generated by `root - 1` to state |
| residue one. This is independent of how a local-ring instance is installed and |
| can be transported directly to a later localization. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace CirculantHadamard.LocalComparison |
|
|
| open CyclicRing Polynomial |
|
|
| variable (A : Type uA) [CommRing A] |
|
|
| / |
| def orderPRoot (p : ℕ) : RamifiedLocal.Extension A p 0 := |
| AdjoinRoot.root (cyclotomic (p ^ (0 + 1)) A) |
|
|
| theorem orderPRoot_pow (p : ℕ) : orderPRoot A p ^ p = 1 := by |
| calc |
| orderPRoot A p ^ p = orderPRoot A p ^ (p ^ (0 + 1)) := |
| congrArg (fun n : ℕ => orderPRoot A p ^ n) (by simp) |
| _ = 1 := cyclotomicAdjoinRoot_pow (R := A) (p ^ (0 + 1)) |
|
|
| theorem orderPRoot_pow_level (p e : ℕ) : orderPRoot A p ^ (p ^ (e + 1)) = 1 := by |
| calc |
| orderPRoot A p ^ (p ^ (e + 1)) = orderPRoot A p ^ (p * p ^ e) := |
| congrArg (fun n : ℕ => orderPRoot A p ^ n) (pow_succ' p e) |
| _ = (orderPRoot A p ^ p) ^ (p ^ e) := pow_mul _ _ _ |
| _ = 1 := by rw [orderPRoot_pow, one_pow] |
| |
| /-- Actual order-`p` evaluation of an element of the cyclic group ring of |
| order `p ^ (e + 1)`. -/ |
| def orderPValue (p e : ℕ) [NeZero p] : |
| Elem A (p ^ (e + 1)) →ₐ[A] RamifiedLocal.Extension A p 0 := |
| evaluateAt (p ^ (e + 1)) (orderPRoot A p) (orderPRoot_pow_level A p e) |
| |
| @[simp] theorem orderPValue_scalar (p e : ℕ) [NeZero p] (a : A) : |
| orderPValue A p e (scalar (p ^ (e + 1)) a) = |
| algebraMap A (RamifiedLocal.Extension A p 0) a := |
| (orderPValue A p e).commutes a |
| |
| theorem orderPValue_projection (p e : ℕ) [NeZero p] |
| (F : Elem A (p ^ (e + 2))) : |
| orderPValue A p e |
| (cyclicProjection A (pow_dvd_pow p (Nat.le_succ (e + 1))) F) = |
| orderPValue A p (e + 1) F := |
| evaluateAt_cyclicProjection _ _ _ _ F |
| |
| /-- The conclusion of comparison, expressed by an actual unit and membership |
| in the actual ramified maximal ideal. -/ |
| def OrderPCompared (p e : ℕ) [NeZero p] (F : Elem A (p ^ (e + 1))) : Prop := |
| ∃ u : (RamifiedLocal.Extension A p 0)ˣ, |
| orderPValue A p e F = (u : RamifiedLocal.Extension A p 0) * |
| algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ (e + 1)) F) ∧ |
| (u : RamifiedLocal.Extension A p 0) - 1 ∈ |
| Ideal.span {RamifiedLocal.uniformizer A p 0} |
| |
| section DVR |
| |
| variable [IsDomain A] [CharZero A] [IsDiscreteValuationRing A] |
| |
| /-- Character comparison with every ramified extension constructed from the |
| input DVR. Both factors are compared, with matching group/scalar exponents. -/ |
| theorem local_character_comparison (p : ℕ) [NeZero p] (hp : p.Prime) |
| (hmax : IsLocalRing.maximalIdeal A = Ideal.span {(p : A)}) |
| (e : ℕ) (b : Aˣ) (F K : Elem A (p ^ (e + 1))) |
| (hFK : F * K = scalar (p ^ (e + 1)) ((p : A) ^ (e + 1) * (b : A))) : |
| OrderPCompared A p e F ∧ OrderPCompared A p e K := by |
| let : IsDomain (RamifiedLocal.Extension A p 0) := |
| RamifiedLocal.isDomain A hp 0 hmax |
| let : IsLocalRing (RamifiedLocal.Extension A p 0) := |
| RamifiedLocal.isLocalRing A hp 0 hmax |
| have hpA : Prime (p : A) := |
| ((IsDiscreteValuationRing.irreducible_iff_uniformizer (p : A)).mpr hmax).prime |
| have hpS : algebraMap A (RamifiedLocal.Extension A p 0) (p : A) ≠ 0 := by |
| intro h |
| exact hpA.ne_zero (RamifiedLocal.algebraMap_injective A hp 0 |
| (by simpa only [map_zero] using h)) |
| have hroot : IsLocalRing.residue (RamifiedLocal.Extension A p 0) (orderPRoot A p) = 1 := by |
| have hpi : RamifiedLocal.uniformizer A p 0 ∈ |
| IsLocalRing.maximalIdeal (RamifiedLocal.Extension A p 0) := by |
| rw [RamifiedLocal.maximalIdeal_eq_span A hp 0 hmax] |
| exact Ideal.subset_span (Set.mem_singleton _) |
| have hz := (IsLocalRing.residue_eq_zero_iff _).mpr hpi |
| simpa only [RamifiedLocal.uniformizer, orderPRoot, map_sub, map_one, sub_eq_zero] using hz |
| have hc := comparison_induction p (algebraMap A (RamifiedLocal.Extension A p 0)) |
| hpA hpS (fun k => (orderPValue A p k).toRingHom) |
| (fun k a => orderPValue_scalar A p k a) |
| (by |
| intro U |
| change IsLocalRing.residue (RamifiedLocal.Extension A p 0) (orderPValue A p 0 U) = |
| IsLocalRing.residue (RamifiedLocal.Extension A p 0) |
| (algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ 1) U)) |
| simp only [orderPValue, evaluateAt_apply, |
| augmentation_apply, map_sum, map_mul, map_pow, hroot, one_pow, mul_one]) |
| (fun k U => orderPValue_projection A p k U) |
| (by |
| intro k c U V hUV |
| let : IsDomain (RamifiedLocal.Extension A p (k + 1)) := |
| RamifiedLocal.isDomain A hp (k + 1) hmax |
| let : IsLocalRing (RamifiedLocal.Extension A p (k + 1)) := |
| RamifiedLocal.isLocalRing A hp (k + 1) hmax |
| let : IsDiscreteValuationRing (RamifiedLocal.Extension A p (k + 1)) := |
| RamifiedLocal.isDiscreteValuationRing A hp (k + 1) hmax |
| have hpR : (p : RamifiedLocal.Extension A p (k + 1)) ≠ 0 := by |
| intro h |
| apply hpA.ne_zero |
| apply RamifiedLocal.algebraMap_injective A hp (k + 1) |
| simpa only [map_natCast, map_zero] using h |
| exact projected_factor_divisible hp hpR |
| (Nat.mul_pos (Nat.sub_pos_of_lt hp.one_lt) (pow_pos hp.pos (k + 1))) |
| (RamifiedLocal.addVal_prime A hp (k + 1) hmax) c U V hUV) |
| e b F K hFK |
| have toCompared {U : Elem A (p ^ (e + 1))} |
| (hU : ResidueOneAssociated (IsLocalRing.residue (RamifiedLocal.Extension A p 0)) |
| (orderPValue A p e U) |
| (algebraMap A (RamifiedLocal.Extension A p 0) (augmentation (p ^ (e + 1)) U))) : |
| OrderPCompared A p e U := by |
| obtain ⟨u, hu, hres⟩ := hU |
| refine ⟨u, hu, ?_⟩ |
| rw [← RamifiedLocal.maximalIdeal_eq_span A hp 0 hmax] |
| apply (IsLocalRing.residue_eq_zero_iff _).mp |
| rw [map_sub, hres, map_one, sub_self] |
| exact ⟨toCompared hc.1, toCompared hc.2⟩ |
| |
| omit [IsDiscreteValuationRing A] in |
| /-- The augmentation denominators used in the corresponding ratio statement |
| are nonzero in the coefficient DVR. -/ |
| theorem local_character_comparison_denominators [IsDiscreteValuationRing A] (p : ℕ) (hp : p.Prime) |
| (e : ℕ) (b : Aˣ) (F K : Elem A (p ^ (e + 1))) |
| (hFK : F * K = scalar (p ^ (e + 1)) ((p : A) ^ (e + 1) * (b : A))) : |
| augmentation (p ^ (e + 1)) F ≠ 0 ∧ augmentation (p ^ (e + 1)) K ≠ 0 := |
| augmentation_ne_zero_of_scalar_product (Nat.cast_ne_zero.mpr hp.ne_zero) b F K hFK |
| |
| end DVR |
| |
| end CirculantHadamard.LocalComparison |
| |
| end |
| |
| end OAI |
| |
