Plain-language statement
E₄.toFun = H₂² + H₂H₄ + H₄². Both are weight-4 level-1 modular forms tending to 1 at ∞, so their difference is a weight-4 cusp form, hence zero.
Exact Lean statement
theorem E₄_eq_H_sum_sq : _root_.E₄.toFun = H_sum_sq
Formal artifact
Lean source
theorem E₄_eq_H_sum_sq : _root_.E₄.toFun = H_sum_sq := by have h_toFun : (_root_.E₄ - H_sum_sq_MF).toFun = _root_.E₄.toFun - H_sum_sq := by ext z; simp [H_sum_sq_MF, H_sum_sq_SIF]; rfl have h_diff_tendsto : Tendsto (_root_.E₄ - H_sum_sq_MF).toFun atImInfty (nhds 0) := by rw [h_toFun] change Tendsto (fun z => _root_.E₄.toFun z - H_sum_sq z) atImInfty (nhds 0) simpa using E₄_tendsto_one_atImInfty.sub H_sum_sq_tendsto have h_cusp : IsCuspForm (Γ 1) 4 (_root_.E₄ - H_sum_sq_MF) := by rw [IsCuspForm_iff_coeffZero_eq_zero, qExpansion_coeff]; simp exact IsZeroAtImInfty.cuspFunction_apply_zero h_diff_tendsto (by norm_num : (0 : ℝ) < 1) have h_zero := IsCuspForm_weight_lt_eq_zero 4 (by norm_num) (_root_.E₄ - H_sum_sq_MF) h_cusp funext z have hz : _root_.E₄.toFun z = (H_sum_sq_MF : ℍ → ℂ) z := by simpa [sub_eq_zero] using DFunLike.congr_fun h_zero z change _root_.E₄.toFun z = H_sum_sq z exact hz- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/JacobiTheta/Derivative.lean:609-624
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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.