| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.HeckeFactors |
|
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| namespace OAI |
|
|
| /-! |
| |
|
|
| The product of norms of conductor primes, raised to their conductor |
| exponents, gives a rational integer imposing every required local |
| higher-unit congruence. No quantitative conductor bound is needed. |
| -/ |
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|
| noncomputable section |
| open scoped NumberField BigOperators Classical |
|
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| namespace DukePrimeDegree |
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| open PrimeDegreePackets |
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| variable {K : Type*} [Field K] [NumberField K] |
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|
| theorem integral_mem_completed_maximal |
| (v : IsDedekindDomain.HeightOneSpectrum (𝓞 K)) |
| {a : 𝓞 K} (ha : a ∈ v.asIdeal) : |
| algebraMap (𝓞 K) (v.adicCompletionIntegers K) a ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) := by |
| apply (Valuation.mem_maximalIdeal_iff (v.adicCompletion K) Valued.v).mpr |
| change Valued.v ((algebraMap (𝓞 K) K a : K) : v.adicCompletion K) < 1 |
| rw [v.valuedAdicCompletion_eq_valuation', v.valuation_of_algebraMap] |
| exact (v.intValuation_lt_one_iff_mem a).mpr ha |
|
|
| theorem exists_integer_for_local_congruences |
| (S : Finset (IsDedekindDomain.HeightOneSpectrum (𝓞 K))) |
| (c : IsDedekindDomain.HeightOneSpectrum (𝓞 K) → ℕ) |
| {D : ℕ} (hD : 0 < D) : |
| ∃ N : ℕ, 0 < N ∧ D ∣ N ∧ ∀ v ∈ S, |
| algebraMap (𝓞 K) (v.adicCompletionIntegers K) (N : 𝓞 K) ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ c v := by |
| let N := D * ∏ v ∈ S, Ideal.absNorm v.asIdeal ^ c v |
| have hnorm (v : IsDedekindDomain.HeightOneSpectrum (𝓞 K)) : |
| 0 < Ideal.absNorm v.asIdeal := by |
| apply Nat.pos_of_ne_zero |
| intro h |
| exact v.ne_bot (Ideal.absNorm_eq_zero_iff.mp h) |
| refine ⟨N, Nat.mul_pos hD (Finset.prod_pos (fun v _ => pow_pos (hnorm v) _)), |
| dvd_mul_right D _, ?_⟩ |
| intro v hv |
| have hd : Ideal.absNorm v.asIdeal ^ c v ∣ N := by |
| exact dvd_mul_of_dvd_right (Finset.dvd_prod_of_mem _ hv) _ |
| obtain ⟨k, hk⟩ := hd |
| have hm := Ideal.pow_mem_pow |
| (integral_mem_completed_maximal v (Ideal.absNorm_mem v.asIdeal)) (c v) |
| rw [hk, Nat.cast_mul, map_mul, Nat.cast_pow, map_pow] |
| exact Ideal.mul_mem_right _ _ hm |
|
|
| theorem completed_congruence_of_integer |
| (v : IsDedekindDomain.HeightOneSpectrum (𝓞 K)) (c N : ℕ) |
| (hN : algebraMap (𝓞 K) (v.adicCompletionIntegers K) (N : 𝓞 K) ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ c) |
| {a : 𝓞 K} (ha : a - 1 ∈ ThetaConstruction.rationalModulusIdeal (𝓞 K) N) : |
| algebraMap (𝓞 K) (v.adicCompletionIntegers K) a - 1 ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ c := by |
| have hd : (N : 𝓞 K) ∣ a - 1 := Ideal.mem_span_singleton.mp ha |
| obtain ⟨b, hb⟩ := hd |
| rw [← map_one (algebraMap (𝓞 K) (v.adicCompletionIntegers K)), ← map_sub, hb, map_mul] |
| exact Ideal.mul_mem_right _ _ hN |
|
|
| theorem completed_integral_isUnit_of_congruence |
| (v : IsDedekindDomain.HeightOneSpectrum (𝓞 K)) {c : ℕ} (hc : 0 < c) |
| {a : 𝓞 K} |
| (ha : algebraMap (𝓞 K) (v.adicCompletionIntegers K) a - 1 ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) ^ c) : |
| IsUnit (algebraMap (𝓞 K) (v.adicCompletionIntegers K) a) := by |
| apply (IsLocalRing.notMem_maximalIdeal).mp |
| intro hmem |
| have hsub := (Ideal.pow_le_self hc.ne') ha |
| have hone : (1 : v.adicCompletionIntegers K) ∈ |
| IsLocalRing.maximalIdeal (v.adicCompletionIntegers K) := by |
| simpa only [sub_sub_cancel] using |
| (IsLocalRing.maximalIdeal (v.adicCompletionIntegers K)).sub_mem hmem hsub |
| exact (IsLocalRing.maximalIdeal.isMaximal (v.adicCompletionIntegers K)).ne_top |
| ((Ideal.eq_top_iff_one _).mpr hone) |
|
|
| end DukePrimeDegree |
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| end |
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| end OAI |
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