Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Ramanujan E₆

ramanujan_E₆'

sphere packing Fourier analysis modular forms ramanujan identities

Plain-language statement

Serre derivative of E₆: serre_D 6 E₆ = - 2⁻¹ * E₄². Uses the dimension argument: 1. serre_D 6 E₆ is weight-8 slash-invariant (by serre_D_slash_invariant) 2. Weight-8 modular forms are 1-dimensional, spanned by E₄² 3. Constant term is -1/2 (from D E₆ → 0, E₂ → 1, E₆ → 1)

Exact Lean statement

theorem ramanujan_E₆' : serre_D 6 E₆.toFun = - 2⁻¹ * E₄.toFun * E₄.toFun

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem ramanujan_E₆' : serre_D 6 E₆.toFun = - 2⁻¹ * E₄.toFun * E₄.toFun := by  let E₄_sq : ModularForm (CongruenceSubgroup.Gamma 1) 8 :=    (by norm_num : (4 : ℤ) + 4 = 8) ▸ E₄.mul E₄  have hE₄_sq_ne : E₄_sq ≠ 0 := fun h => E4_ne_zero <| by    ext z    have := congrFun (congrArg (↑· : ModularForm _ _ → ℍ → ℂ) h) z    simp at this    exact mul_self_eq_zero.mp this  obtain ⟨c, hc⟩ := exists_smul_eq_of_rank_one    (weight_eight_one_dimensional 8 (by norm_num) ⟨4, rfl⟩ (by norm_num)) hE₄_sq_ne    serre_DE₆_ModularForm  have hfun : ∀ z, serre_D 6 E₆.toFun z = c * (E₄.toFun z * E₄.toFun z) := fun z => by    calc      serre_D 6 E₆.toFun z = (serre_DE₆_ModularForm : ℍ → ℂ) z := rfl      _ = c * (E₄_sq : ℍ → ℂ) z := smul_modularForm_eq_pointwise hc z      _ = c * (E₄.toFun z * E₄.toFun z) := by        congr 1  have hc_val : c = -(1/2 : ℂ) := scalar_eq_of_tendsto hfun serre_DE₆_tendsto_atImInfty    (by tendsto_cont [E₄_tendsto_one_atImInfty])  ext z  simp only [hfun z, hc_val, Pi.mul_apply]  ring_nf  norm_num

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/RamanujanIdentities.lean:105-127

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Project-declaredLean 4.31.0

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sphere packingFourier analysismodular forms

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Person-level attribution pending.

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Project-declaredLean 4.31.0

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Person-level attribution pending.

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Project-declaredLean 4.31.0

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Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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