Plain-language statement
g is invariant under S. Proof: g = (2H₂ + H₄)f₂ + (H₂ + 2H₄)f₄ Under S: H₂ ↦ -H₄, H₄ ↦ -H₂, f₂ ↦ -f₄, f₄ ↦ -f₂ g|S = (2(-H₄) + (-H₂))(-f₄) + ((-H₄) + 2(-H₂))(-f₂) = (2H₄ + H₂)f₄ + (H₄ + 2H₂)f₂ = g
Exact Lean statement
lemma theta_g_S_action : (theta_g ∣[(6 : ℤ)] S) = theta_g
Formal artifact
Lean source
lemma theta_g_S_action : (theta_g ∣[(6 : ℤ)] S) = theta_g := by -- Linear combination transforms: (2•H₂ + H₄)|S = -(2•H₄ + H₂), (H₂ + 2•H₄)|S = -(H₄ + 2•H₂) have h_2H₂_H₄ : (((2 : ℂ) • H₂ + H₄) ∣[(2 : ℤ)] S) = -((2 : ℂ) • H₄ + H₂) := by simp only [add_slash, SL_smul_slash, H₂_S_action, H₄_S_action] ext z; simp [Pi.add_apply, Pi.smul_apply, Pi.neg_apply]; ring have h_H₂_2H₄ : ((H₂ + (2 : ℂ) • H₄) ∣[(2 : ℤ)] S) = -(H₄ + (2 : ℂ) • H₂) := by simp only [add_slash, SL_smul_slash, H₂_S_action, H₄_S_action] ext z; simp [Pi.add_apply, Pi.smul_apply, Pi.neg_apply]; ring -- Product transforms using mul_slash_SL2 have h_term1 : ((((2 : ℂ) • H₂ + H₄) * f₂) ∣[(6 : ℤ)] S) = ((2 : ℂ) • H₄ + H₂) * f₄ := by have hmul := mul_slash_SL2 2 4 S ((2 : ℂ) • H₂ + H₄) f₂ simp only [h_2H₂_H₄, f₂_S_action] at hmul rw [show (2 : ℤ) + 4 = 6 by norm_num] at hmul refine hmul.trans ?_ ext z; simp [Pi.mul_apply, Pi.add_apply, Pi.smul_apply, Pi.neg_apply, smul_eq_mul]; ring have h_term2 : (((H₂ + (2 : ℂ) • H₄) * f₄) ∣[(6 : ℤ)] S) = (H₄ + (2 : ℂ) • H₂) * f₂ := by have hmul := mul_slash_SL2 2 4 S (H₂ + (2 : ℂ) • H₄) f₄ simp only [h_H₂_2H₄, f₄_S_action] at hmul rw [show (2 : ℤ) + 4 = 6 by norm_num] at hmul refine hmul.trans ?_ ext z; simp [Pi.mul_apply, Pi.add_apply, Pi.smul_apply, Pi.neg_apply, smul_eq_mul]; ring -- g|S = (2H₄ + H₂)f₄ + (H₄ + 2H₂)f₂ = g simp only [theta_g, add_slash, h_term1, h_term2] ext z; simp only [Pi.add_apply, Pi.mul_apply, Pi.smul_apply]; ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/JacobiTheta/Derivative.lean:291-318
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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.