Plain-language statement
E₂ transforms under S as: E₂(-1/z) = z² · (E₂(z) + 6/(πIz)).
Exact Lean statement
lemma E₂_S_transform (z : ℍ) :
E₂ (ModularGroup.S • z) = z ^ 2 * (E₂ z + 6 / (π * Complex.I * z))Formal artifact
Lean source
lemma E₂_S_transform (z : ℍ) : E₂ (ModularGroup.S • z) = z ^ 2 * (E₂ z + 6 / (π * Complex.I * z)) := by have h := E₂_transform z rw [SL_slash_apply, ModularGroup.denom_S, zpow_neg, zpow_two] at h have hz2 : (z : ℂ) * (z : ℂ) ≠ 0 := mul_ne_zero (ne_zero z) (ne_zero z) rw [sq, mul_comm] -- `only` is required here; without it simp rewrites the congrArg term structure simpa only [mul_assoc, inv_mul_cancel₀ hz2, mul_one] using congrArg (· * ((z : ℂ) * (z : ℂ))) h- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/E2.lean:77-84
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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.