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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 | import OAI.NumberTheory.DukePrimeDegree.Arithmetic.LatticeBasis
import OAI.NumberTheory.DukePrimeDegree.Geometry.EmbeddingMatrix
import OAI.NumberTheory.PrimePackets.PacketAlgebra
namespace OAI
/-!
# Full lattices in the existing packet model
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
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