| import Mathlib |
| import OAI.NumberTheory.PiExponent.Geometry.CurveLocalOrder |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped nonZeroDivisors |
| open PiExponent.WeightedCurveDegree PiExponent.CurveLocalOrder |
|
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| namespace PiExponent.DVRValuationUnique |
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| variable {A E : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] |
| [Field E] [Algebra A E] [IsFractionRing A E] |
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| def localFieldUnit (a : A) (ha : a ≠ 0) : Eˣ := |
| Units.mk0 (algebraMap A E a) ((map_ne_zero_iff _ (IsFractionRing.injective A E)).mpr ha) |
|
|
| omit [IsDomain A] [IsDiscreteValuationRing A] in |
| @[simp] theorem localFieldUnit_val (a : A) (ha : a ≠ 0) : |
| ((localFieldUnit (E := E) a ha : Eˣ) : E) = algebraMap A E a := rfl |
|
|
| omit [IsDomain A] [IsDiscreteValuationRing A] [IsFractionRing A E] in |
| theorem integerOrder_unit_eq_zero |
| (v : AddValuation E (WithTop ℤ)) |
| (hregular : ∀ a : A, 0 ≤ v (algebraMap A E a)) (u : Aˣ) : |
| integerOrder v (Units.map (algebraMap A E) u) = 0 := by |
| have hpos : 0 ≤ integerOrder v (Units.map (algebraMap A E) u) := by |
| apply WithTop.coe_le_coe.mp |
| simpa using hregular (u : A) |
| have hneg : 0 ≤ integerOrder v (Units.map (algebraMap A E) u⁻¹) := by |
| apply WithTop.coe_le_coe.mp |
| simpa using hregular (u⁻¹ : Aˣ) |
| simp only [map_inv, integerOrder_inv] at hneg |
| omega |
|
|
| theorem exists_order_eq_uniformizer_multiple |
| (v : AddValuation E (WithTop ℤ)) |
| (hregular : ∀ a : A, 0 ≤ v (algebraMap A E a)) |
| {π : A} (hπ : Irreducible π) (a : A) (ha : a ≠ 0) : |
| ∃ n : ℕ, integerOrder v (localFieldUnit (E := E) a ha) = |
| (n : ℤ) * integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) ∧ |
| IsDiscreteValuationRing.addVal A a = n := by |
| obtain ⟨n, u, rfl⟩ := IsDiscreteValuationRing.eq_unit_mul_pow_irreducible ha hπ |
| refine ⟨n, ?_, IsDiscreteValuationRing.addVal_def' u hπ n⟩ |
| have he : localFieldUnit (E := E) ((u : A) * π ^ n) ha = |
| Units.map (algebraMap A E) u * localFieldUnit (E := E) π hπ.ne_zero ^ n := by |
| ext |
| simp [localFieldUnit, map_mul, map_pow] |
| rw [he, integerOrder_mul, integerOrder_pow, integerOrder_unit_eq_zero v hregular] |
| simp |
|
|
| theorem uniformizer_order_dvd |
| (v : AddValuation E (WithTop ℤ)) |
| (hregular : ∀ a : A, 0 ≤ v (algebraMap A E a)) |
| {π : A} (hπ : Irreducible π) (z : Eˣ) : |
| integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) ∣ integerOrder v z := by |
| obtain ⟨a, b, hb, hab⟩ := IsFractionRing.div_surjective A (z : E) |
| have hb0 : b ≠ 0 := mem_nonZeroDivisors_iff_ne_zero.mp hb |
| have ha0 : a ≠ 0 := by |
| intro ha |
| apply z.ne_zero |
| rw [← hab, ha, map_zero, zero_div] |
| have hz : z = localFieldUnit (E := E) a ha0 / localFieldUnit (E := E) b hb0 := by |
| ext |
| simpa only [Units.val_div_eq_div_val, localFieldUnit_val] using hab.symm |
| obtain ⟨m, hm, _⟩ := exists_order_eq_uniformizer_multiple v hregular hπ a ha0 |
| obtain ⟨n, hn, _⟩ := exists_order_eq_uniformizer_multiple v hregular hπ b hb0 |
| rw [hz, integerOrder_div, hm, hn] |
| exact ⟨(m : ℤ) - n, by ring⟩ |
|
|
| theorem eq_fractionAddValuation |
| (v : AddValuation E (WithTop ℤ)) |
| (hregular : ∀ a : A, 0 ≤ v (algebraMap A E a)) |
| (hcenter : ∀ a ∈ IsLocalRing.maximalIdeal A, a ≠ 0 → 0 < v (algebraMap A E a)) |
| (hnormalized : ∃ z : Eˣ, integerOrder v z = 1) : |
| v = fractionAddValuation A E := by |
| obtain ⟨π, hπ⟩ := IsDiscreteValuationRing.exists_irreducible A |
| have hπmem : π ∈ IsLocalRing.maximalIdeal A := by |
| rw [hπ.maximalIdeal_eq] |
| exact Ideal.subset_span (Set.mem_singleton π) |
| have hpos : 0 < integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) := by |
| apply WithTop.coe_lt_coe.mp |
| simpa using hcenter π hπmem hπ.ne_zero |
| have hone : integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) = 1 := by |
| obtain ⟨z, hz⟩ := hnormalized |
| apply Int.eq_one_of_dvd_one hpos.le |
| rw [← hz] |
| exact uniformizer_order_dvd v hregular hπ z |
| have hlocal (a : A) : v (algebraMap A E a) = localAddValuation A a := by |
| by_cases ha : a = 0 |
| · simp [ha] |
| obtain ⟨n, hn, hcanonical⟩ := exists_order_eq_uniformizer_multiple v hregular hπ a ha |
| rw [hone, mul_one] at hn |
| rw [localAddValuation_apply, hcanonical, enatToIntegerOrder_natCast] |
| rw [← localFieldUnit_val a ha, ← coe_integerOrder, hn] |
| rfl |
| ext x |
| obtain ⟨a, b, _, rfl⟩ := IsFractionRing.div_surjective A x |
| rw [AddValuation.map_div, AddValuation.map_div, hlocal, hlocal, |
| fractionAddValuation_algebraMap, fractionAddValuation_algebraMap] |
|
|
| theorem fractionAddValuation_normalized : |
| ∃ z : Eˣ, integerOrder (fractionAddValuation A E) z = 1 := by |
| obtain ⟨π, hπ⟩ := IsDiscreteValuationRing.exists_irreducible A |
| refine ⟨localFieldUnit (E := E) π hπ.ne_zero, ?_⟩ |
| apply WithTop.coe_injective |
| rw [coe_integerOrder, localFieldUnit_val, fractionAddValuation_algebraMap, |
| localAddValuation_apply, IsDiscreteValuationRing.addVal_uniformizer hπ] |
| rfl |
|
|
| end PiExponent.DVRValuationUnique |
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| end |
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| end OAI |
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