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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 | 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
/-- Conditional form of Theorem 1.1, including tightness of the entire
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
/-- The same sequence converges to the distinguished invariant
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
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