Ramanujan Identities for Eisenstein Series

This file contains the fundamental Ramanujan identities for Eisenstein series (Blueprint Theorem 6.50).

Main results (Serre derivative forms)

• ramanujan_E₂' : serre_D 1 E₂ = -E₄/12 (requires explicit computation since E₂ is not modular)
• ramanujan_E₄' : serre_D 4 E₄ = -E₆/3 (uses dimension formula for weight-6 forms)
• ramanujan_E₆' : serre_D 6 E₆ = -E₄²/2 (uses dimension formula for weight-8 forms)

Derived identities (D-derivative forms)

• ramanujan_E₂ : D E₂ = (E₂² - E₄)/12
• ramanujan_E₄ : D E₄ = (E₂·E₄ - E₆)/3
• ramanujan_E₆ : D E₆ = (E₂·E₆ - E₄²)/2

Proof Strategy

Uses dimension formulas: dim M_k(Γ(1)) = 1 for k = 4, 6, 8. Since serre_D k E_k is a modular form in the 1-dimensional space, it must be a scalar multiple of the unique generator. The scalar is determined by comparing limits as z → i∞.

The Ramanujan Identities

These are the main theorems. The primed versions are in terms of serre_D, the non-primed versions are in terms of D.

theorem scalar_eq_of_tendsto {f g : UpperHalfPlane → ℂ} {c L : ℂ} (hfun : ∀ (z : UpperHalfPlane), f z = c * g z) (hf_lim : Filter.Tendsto f UpperHalfPlane.atImInfty (nhds L)) (hg_lim : Filter.Tendsto g UpperHalfPlane.atImInfty (nhds 1)) : c = L

Determine scalar coefficient from limits: if f = c * g pointwise, f → L at i∞, and g → 1 at i∞, then c = L.

This captures the "uniqueness of limits" argument used in dimension-1 proofs.

theorem ramanujan_E₂' : serre_D 1 E₂ = -12⁻¹ * E₄.toFun

Serre derivative of E₂: serre_D 1 E₂ = - 12⁻¹ * E₄.

This is the hardest identity because E₂ is not modular. The proof uses:

1. serre_DE₂_slash_invariant: serre_D 1 E₂ is weight-4 slash-invariant
2. Bounded at infinity: serre_D 1 E₂ = D E₂ - (1/12) E₂², both terms bounded
3. Dimension formula: weight-4 forms are 1-dimensional, spanned by E₄
4. Constant term: serre_D 1 E₂(iy) → -1/12 as y → ∞

theorem ramanujan_E₄' : serre_D 4 E₄.toFun = -3⁻¹ * E₆.toFun

Serre derivative of E₄: serre_D 4 E₄ = - 3⁻¹ * E₆.

Uses the dimension argument:

1. serre_D 4 E₄ is weight-6 slash-invariant (by serre_D_slash_invariant)
2. serre_D 4 E₄ is bounded at infinity (serre_DE₄_isBoundedAtImInfty)
3. Weight-6 modular forms are 1-dimensional (weight_six_one_dimensional)
4. Constant term is -1/3 (from D E₄ → 0, E₂ → 1, E₄ → 1)

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

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)

Derived Ramanujan identities (D instead of serre_D)

theorem D_eq_serre_D_add (k : ℂ) (f : UpperHalfPlane → ℂ) (z : UpperHalfPlane) : D f z = serre_D k f z + k * 12⁻¹ * E₂ z * f z

Relationship between D and serre_D: D f = serre_D k f + k/12 * E₂ * f.

@[simp]

theorem ramanujan_E₂ : D E₂ = 12⁻¹ * (E₂ * E₂ - E₄.toFun)

@[simp]

theorem ramanujan_E₄ : D E₄.toFun = 3⁻¹ * (E₂ * E₄.toFun - E₆.toFun)

@[simp]

theorem ramanujan_E₆ : D E₆.toFun = 2⁻¹ * (E₂ * E₆.toFun - E₄.toFun * E₄.toFun)