| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.LatticeBasis |
| import OAI.NumberTheory.DukePrimeDegree.Geometry.EmbeddingMatrix |
| import OAI.NumberTheory.PrimePackets.PacketAlgebra |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| The packet development records a chosen integer basis in its full-lattice |
| structure. Our finite-generation definition produces that basis. This |
| conversion preserves the actual submodule and multiplier order. |
| -/ |
|
|
| namespace DukePrimeDegree |
|
|
| variable {K : Type*} [Field K] [NumberField K] {n : ℕ} |
|
|
| /-- Equip a full lattice with the basis expected by the packet model. -/ |
| noncomputable def FullLattice.toPacket (M : FullLattice K) |
| (hn : Module.finrank ℚ K = n) : PrimeDegreePackets.FullLattice n K where |
| carrier := M.toSubmodule |
| basis := M.integralBasis.reindex (finCongr hn) |
| spans := M.span_eq_top |
|
|
| @[simp] lemma FullLattice.toPacket_carrier (M : FullLattice K) |
| (hn : Module.finrank ℚ K = n) : (M.toPacket hn).carrier = M.toSubmodule := rfl |
|
|
| @[simp] lemma FullLattice.toPacket_multiplierOrder (M : FullLattice K) |
| (hn : Module.finrank ℚ K = n) : |
| PrimeDegreePackets.multiplierOrder (M.toPacket hn) = |
| (multiplierOrder M.toSubmodule).toSubring.toAddSubgroup := by |
| ext x |
| rfl |
|
|
| /-- Convert the complex-embedding ordering to the real-embedding ordering |
| used by the row-lattice packet model. Total reality makes this bijective. -/ |
| noncomputable def EmbeddingOrder.toRealOrder [NumberField.IsTotallyReal K] |
| (σ : EmbeddingOrder K n) : PrimeDegreePackets.OrderedEmbeddings n K := |
| Equiv.ofBijective (orderedRealEmbedding σ) (by |
| constructor |
| · intro i j hij |
| apply σ.injective |
| apply AlgHom.ext |
| intro x |
| have h := congrArg (fun f : K →+* ℝ => (f x : ℂ)) hij |
| simpa only [orderedRealEmbedding_complex] using h |
| · intro τ |
| let τc : K →ₐ[ℚ] ℂ := (Complex.ofRealHom.comp τ).toRatAlgHom |
| obtain ⟨i, hi⟩ := σ.surjective τc |
| refine ⟨i, ?_⟩ |
| ext x |
| apply Complex.ofReal_injective |
| rw [orderedRealEmbedding_complex, hi] |
| rfl) |
|
|
| @[simp] lemma EmbeddingOrder.toRealOrder_apply [NumberField.IsTotallyReal K] |
| (σ : EmbeddingOrder K n) (i : Fin n) : σ.toRealOrder i = orderedRealEmbedding σ i := rfl |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
|
|
