Plain-language statement
The D-derivative of the anomaly function D₂. D₂ γ z = 2πi · (γ₁₀ / denom γ z), so D(D₂ γ) = (2πi)⁻¹ · d/dz[2πi · c / denom] = -c² / denom²
Exact Lean statement
lemma D_D₂ (γ : SL(2, ℤ)) (z : ℍ) :
D (D₂ γ) z = - (γ 1 0 : ℂ)^2 / (denom γ z)^2Formal artifact
Lean source
lemma D_D₂ (γ : SL(2, ℤ)) (z : ℍ) : D (D₂ γ) z = - (γ 1 0 : ℂ)^2 / (denom γ z)^2 := by have hz_ne : denom γ z ≠ 0 := UpperHalfPlane.denom_ne_zero γ z have hderiv : deriv ((D₂ γ) ∘ ofComplex) z = deriv (fun w => (2 * π * I * (γ 1 0 : ℂ)) / denom γ w) z := by apply Filter.EventuallyEq.deriv_eq filter_upwards [isOpen_upperHalfPlaneSet.mem_nhds z.im_pos] with w hw simp only [comp_apply, ofComplex_apply_of_im_pos hw, D₂, EisensteinSeries.D2] simp only [D, hderiv, div_eq_mul_inv, ← zpow_neg_one] rw [deriv_const_mul _ (.zpow (differentiableAt_denom γ z) (.inl hz_ne)), deriv_denom_zpow γ 1 z] simp only [Int.reduceNeg, Int.reduceSub, zpow_neg_one]; field_simp; ring- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/SerreDerivativeSlash.lean:41-52
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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.