hpOperator_eigenvector_pos

formal source

  65roots `±θ`.  We exhibit eigenvectors directly. -/
  66
  67/-- The vector `(1, 1)` is an eigenvector with eigenvalue `+θ`. -/
  68theorem hpOperator_eigenvector_pos (q : ℕ) (a : ℤ) :
  69    (hpOperator q a).mulVec ![1, 1] = frobeniusAngle q a • ![1, 1] := by
  70  ext i
  71  fin_cases i <;>
  72    simp [hpOperator, Matrix.mulVec, Matrix.cons_val', Matrix.empty_val',
  73          Matrix.cons_val_fin_one, Matrix.cons_val_zero, Matrix.cons_val_one,
  74          Matrix.head_cons, Matrix.head_fin_const,
  75          Matrix.cons_dotProduct, Matrix.dotProduct_empty, Fin.sum_univ_two] <;>
  76    ring
  77
  78/-- The vector `(1, -1)` is an eigenvector with eigenvalue `-θ`. -/
  79theorem hpOperator_eigenvector_neg (q : ℕ) (a : ℤ) :
  80    (hpOperator q a).mulVec ![1, -1] = (-frobeniusAngle q a) • ![1, -1] := by
  81  ext i
  82  fin_cases i <;>
  83    simp [hpOperator, Matrix.mulVec, Matrix.cons_val', Matrix.empty_val',
  84          Matrix.cons_val_fin_one, Matrix.cons_val_zero, Matrix.cons_val_one,
  85          Matrix.head_cons, Matrix.head_fin_const,
  86          Matrix.cons_dotProduct, Matrix.dotProduct_empty, Fin.sum_univ_two] <;>
  87    ring
  88
  89/-! ## Hasse bound implies real spectrum
  90
  91For the angle `θ` to be a real number representing a meaningful spectral
  92quantity, we need `arccos`'s argument to lie in `[-1, 1]`.  This is the
  93content of the Hasse--Weil bound. -/
  94
  95/-- If the Hasse bound `a^2 ≤ 4q` holds, then `a / (2√q) ∈ [-1, 1]`. -/
  96theorem hasse_implies_arccos_valid
  97    (q : ℕ) (hq : 0 < q) (a : ℤ) (h_hasse : hasseBound q a) :
  98    ((a : ℝ) / (2 * Real.sqrt (q : ℝ))) ∈ Set.Icc (-1 : ℝ) 1 := by