Plain-language statement
h is invariant under T. Proof: Under T: f₂ ↦ -f₂, f₄ ↦ f₃ = f₂ + f₄ h|T = (-f₂)² + (-f₂)(f₂ + f₄) + (f₂ + f₄)² = f₂² - f₂² - f₂f₄ + f₂² + 2f₂f₄ + f₄² = f₂² + f₂f₄ + f₄² = h
Exact Lean statement
lemma theta_h_T_action : (theta_h ∣[(8 : ℤ)] T) = theta_h
Formal artifact
Lean source
lemma theta_h_T_action : (theta_h ∣[(8 : ℤ)] T) = theta_h := by -- Under T: f₂ ↦ -f₂, f₄ ↦ f₃ = f₂ + f₄ -- (f₂²)|T = f₂², (f₄²)|T = (f₂+f₄)², (f₂f₄)|T = (-f₂)(f₂+f₄) have h_f₂_sq : ((f₂ ^ 2) ∣[(8 : ℤ)] T) = f₂ ^ 2 := by have hmul := mul_slash_SL2 4 4 T f₂ f₂ simp only [f₂_T_action] at hmul convert hmul using 1 <;> ext <;> simp [sq] have h_f₄_sq : ((f₄ ^ 2) ∣[(8 : ℤ)] T) = (f₂ + f₄) ^ 2 := by have hmul := mul_slash_SL2 4 4 T f₄ f₄ simp only [f₄_T_action] at hmul convert hmul using 1 · ext; simp [sq] · ext z; simp only [Pi.pow_apply, Pi.mul_apply, sq] rw [(congrFun f₂_add_f₄_eq_f₃ z).symm, Pi.add_apply] have h_f₂f₄ : ((f₂ * f₄) ∣[(8 : ℤ)] T) = (-f₂) * (f₂ + f₄) := by have hmul := mul_slash_SL2 4 4 T f₂ f₄ simp only [f₂_T_action, f₄_T_action] at hmul simpa [show (4 : ℤ) + 4 = 8 by norm_num, ← f₂_add_f₄_eq_f₃] using hmul -- h|T = f₂² + (-f₂)(f₂+f₄) + (f₂+f₄)² = h simp only [theta_h, add_slash, h_f₂_sq, h_f₂f₄, h_f₄_sq] ext z simp only [Pi.add_apply, Pi.mul_apply, Pi.neg_apply, sq] ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/JacobiTheta/Derivative.lean:385-407
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.