File size: 3,222 Bytes
a44267b | 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 | import OAI.NumberTheory.DukePrimeDegree.Arithmetic.LatticeBasis
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.RingTheory.Discriminant
namespace OAI
/-!
# Ordered real embeddings and lattice basis matrices
An ordering of the complex embeddings of a totally real field gives the
ordered real embeddings used in the manuscript. The matrix of any full
lattice basis under these embeddings has nonzero determinant.
-/
namespace DukePrimeDegree
variable {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyReal K]
{n : ℕ}
/-- An ordering of all embeddings; total reality makes their values real. -/
abbrev EmbeddingOrder (K : Type*) [Field K] [NumberField K] (n : ℕ) :=
Fin n ≃ (K →ₐ[ℚ] ℂ)
/-- The real ring embedding at one place of the ordering. -/
noncomputable def orderedRealEmbedding (σ : EmbeddingOrder K n) (j : Fin n) : K →+* ℝ :=
(NumberField.IsTotallyReal.complexEmbedding_isReal (σ j).toRingHom).embedding
@[simp] lemma orderedRealEmbedding_complex (σ : EmbeddingOrder K n)
(j : Fin n) (x : K) : (orderedRealEmbedding σ j x : ℂ) = σ j x :=
NumberField.ComplexEmbedding.IsReal.coe_embedding_apply _ _
/-- Rows are basis vectors and columns are ordered real embeddings. -/
noncomputable def realEmbeddingMatrix (b : Fin n → K) (σ : EmbeddingOrder K n) :
Matrix (Fin n) (Fin n) ℝ := fun i j => orderedRealEmbedding σ j (b i)
lemma realEmbeddingMatrix_complex (b : Fin n → K) (σ : EmbeddingOrder K n) :
(realEmbeddingMatrix b σ).map Complex.ofRealHom =
Algebra.embeddingsMatrixReindex ℚ ℂ b σ := by
ext i j
simp [realEmbeddingMatrix, Algebra.embeddingsMatrixReindex, Algebra.embeddingsMatrix]
/-- The field trace discriminant proves the embedded lattice basis is
linearly independent over the real numbers. -/
theorem det_realEmbeddingMatrix_ne_zero (b : Module.Basis (Fin n) ℚ K)
(σ : EmbeddingOrder K n) : (realEmbeddingMatrix b σ).det ≠ 0 := by
have hdisc : (algebraMap ℚ ℂ) (Algebra.discr ℚ b) ≠ 0 := by
intro h
apply Algebra.discr_not_zero_of_basis ℚ b
apply (algebraMap ℚ ℂ).injective
simpa only [map_zero] using h
have heq := Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two ℚ ℂ b σ
have hdet : (Algebra.embeddingsMatrixReindex ℚ ℂ b σ).det ≠ 0 := by
intro h
apply hdisc
rw [heq, h, zero_pow (by decide : (2 : ℕ) ≠ 0)]
intro hreal
apply hdet
rw [← realEmbeddingMatrix_complex]
change (Complex.ofRealHom.mapMatrix (realEmbeddingMatrix b σ)).det = 0
rw [← RingHom.map_det, hreal, map_zero]
/-- In particular this applies to the chosen integer basis of a full lattice. -/
theorem FullLattice.det_embeddingMatrix_ne_zero (M : FullLattice K)
(σ : EmbeddingOrder K (Module.finrank ℚ K)) :
(realEmbeddingMatrix (fun i => (M.integralBasis i : K)) σ).det ≠ 0 := by
have h := det_realEmbeddingMatrix_ne_zero (M.rationalBasis M.integralBasis) σ
have hb : (M.rationalBasis M.integralBasis : Fin (Module.finrank ℚ K) → K) =
fun i => (M.integralBasis i : K) := by
funext i
exact M.rationalBasis_apply M.integralBasis i
rwa [hb] at h
end DukePrimeDegree
end OAI
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