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b3ddf75 | 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 | import OAI.NumberTheory.DukePrimeDegree.Arithmetic.HeckeFactors
namespace OAI
/-!
# An integer modulus for finitely many local congruences
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.
-/
noncomputable section
open scoped NumberField BigOperators Classical
namespace DukePrimeDegree
open PrimeDegreePackets
variable {K : Type*} [Field K] [NumberField K]
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
end
end OAI
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