Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Theta h S action

theta_h_S_action

sphere packing Fourier analysis modular forms derivative

Plain-language statement

h is invariant under S. Proof: h = f₂² + f₂f₄ + f₄² Under S: f₂|[4]S = -f₄, f₄|[4]S = -f₂ Using mul_slash_SL2: (f₂²)|[8]S = (f₂|[4]S)² = (-f₄)² = f₄² (f₂f₄)|[8]S = (f₂|[4]S)(f₄|[4]S) = (-f₄)(-f₂) = f₂f₄ (f₄²)|[8]S = (f₄|[4]S)² = (-f₂)² = f₂² So h|[8]S = f₄² + f₂f₄ + f₂² = f₂² + f₂f₄ + f₄² = h

Exact Lean statement

lemma theta_h_S_action : (theta_h ∣[(8 : ℤ)] S) = theta_h

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma theta_h_S_action : (theta_h ∣[(8 : ℤ)] S) = theta_h := by  -- Under S: f₂ ↦ -f₄, f₄ ↦ -f₂  -- (f₂²)|S = f₄², (f₄²)|S = f₂², (f₂f₄)|S = f₂f₄  have h_f₂_sq : ((f₂ ^ 2) ∣[(8 : ℤ)] S) = f₄ ^ 2 := by    have hmul := mul_slash_SL2 4 4 S f₂ f₂    simp only [f₂_S_action] at hmul    convert hmul using 1 <;> ext <;> simp [sq]  have h_f₄_sq : ((f₄ ^ 2) ∣[(8 : ℤ)] S) = f₂ ^ 2 := by    have hmul := mul_slash_SL2 4 4 S f₄ f₄    simp only [f₄_S_action] at hmul    convert hmul using 1 <;> ext <;> simp [sq]  have h_f₂f₄ : ((f₂ * f₄) ∣[(8 : ℤ)] S) = f₂ * f₄ := by    have hmul := mul_slash_SL2 4 4 S f₂ f₄    simp only [f₂_S_action, f₄_S_action] at hmul    simpa [show (4 : ℤ) + 4 = 8 by norm_num, mul_comm] using hmul  -- h|S = f₄² + f₂f₄ + f₂² = 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, sq]  ring

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/JacobiTheta/Derivative.lean:358-377

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Project-declaredLean 4.31.0

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sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

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Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$ . Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$ , then $F(it)$ is positive for all $t > 0$ .

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Person-level attribution pending.

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Project-declaredLean 4.31.0

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closedBall_center_subset_upperHalfPlane

Plain-language statement

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Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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