| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.PublishedStark |
| import OAI.NumberTheory.DukePrimeDegree.MainConditional |
|
|
| namespace OAI |
|
|
| /-! |
| # Theorem 1.1 conditional on published inputs |
|
|
| This is the manuscript's main result for arbitrary full-lattice local |
| types, including varying number fields and nonmaximal multiplier orders. |
| The only explicit inputs are `StarkInput` (Stark 1974, Lemmas 3 and 8 and |
| equation (9)) and `EKLPrimeInput` (EKL 2006, Theorem 1.3/Corollary 1.4). |
| The intermediate uniform zeta estimate is derived, not assumed. |
| -/ |
|
|
| open MeasureTheory Filter |
| open scoped Topology |
|
|
| namespace DukePrimeDegree |
|
|
| open PrimeDegreePackets |
|
|
| universe u |
|
|
| / |
| sequence of complete volume-weighted local-homothety packets. -/ |
| theorem prime_degree_packet_equidistribution |
| (d : ℕ) (hprime : Nat.Prime (d + 1)) (hfive : 5 ≤ d + 1) |
| (stark : StarkInput.{u} (d + 1)) (ekl : EKLPrimeInput d) |
| (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)] |
| [∀ i, NumberField.IsTotallyReal (K i)] |
| (M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i)) |
| (σ : ∀ i, OrderedEmbeddings (d + 1) (K i)) |
| (hn : ∀ i, Module.finrank ℚ (K i) = d + 1) |
| (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) : |
| ∃ ν : ProbabilityMeasure (X (d + 1)), |
| IsHaarProbability (ν : Measure (X (d + 1))) ∧ |
| Tendsto (fun i => packetProbability (M i) (σ i) (hn i) |
| (hprime.odd_of_ne_two (by omega))) atTop (𝓝 ν) ∧ |
| IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by |
| exact prime_degree_packet_equidistribution_of_scale_and_EKL d hprime hfive |
| (uniformZetaScaleBound_of_stark hprime (by omega) stark) ekl K M σ hn hD |
|
|
| / |
| probability, whenever it is presented explicitly. -/ |
| theorem tendsto_haar_of_published_inputs |
| (d : ℕ) (hprime : Nat.Prime (d + 1)) (hfive : 5 ≤ d + 1) |
| (stark : StarkInput.{u} (d + 1)) (ekl : EKLPrimeInput d) |
| (K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)] |
| [∀ i, NumberField.IsTotallyReal (K i)] |
| (M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i)) |
| (σ : ∀ i, OrderedEmbeddings (d + 1) (K i)) |
| (hn : ∀ i, Module.finrank ℚ (K i) = d + 1) |
| (hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) |
| (m : ProbabilityMeasure (X (d + 1))) (hm : IsHaarProbability (m : Measure (X (d + 1)))) : |
| Tendsto (fun i => packetProbability (M i) (σ i) (hn i) |
| (hprime.odd_of_ne_two (by omega))) atTop (𝓝 m) := by |
| obtain ⟨ν, hν, hconv, _⟩ := |
| prime_degree_packet_equidistribution d hprime hfive stark ekl K M σ hn hD |
| have heq : ν = m := Subtype.ext (isHaarProbability_unique hν hm) |
| rwa [heq] at hconv |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
|
|
