Plain-language statement
The vanishing order of F at infinity is 2. Blueprint: F = 720² * q² * (1 + O(q)), so F / q² → 720² as im(z) → ∞.
Exact Lean statement
theorem F_vanishing_order :
Tendsto (fun z : ℍ ↦ F z / cexp (2 * π * Complex.I * 2 * z))
atImInfty (nhds (720 ^ 2))Formal artifact
Lean source
theorem F_vanishing_order : Tendsto (fun z : ℍ ↦ F z / cexp (2 * π * Complex.I * 2 * z)) atImInfty (nhds (720 ^ 2)) := by have h_exp_eq : ∀ z : ℍ, cexp (2 * π * I * 2 * z) = cexp (2 * π * I * z) ^ 2 := by intro z rw [← Complex.exp_nat_mul] congr 1 ring have h_F_eq : ∀ z : ℍ, F z / cexp (2 * π * I * 2 * z) = ((E₂ z * E₄ z - E₆ z) / cexp (2 * π * I * z)) ^ 2 := by intro z simp only [F, h_exp_eq, sq, div_mul_div_comm, Pi.mul_apply, Pi.sub_apply, ModularForm.toFun_eq_coe] simp_rw [h_F_eq] exact E₂E₄_sub_E₆_div_q_tendsto.pow 2- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:794-808
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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.
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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.