Transformation Rules for φ₀

This file proves the transformation properties of φ₀ under the modular group action, as stated in Blueprint Lemma 7.2:

1. T-periodicity: φ₀(z + 1) = φ₀(z)
2. S-transformation: φ₀(-1/z) = φ₀(z) - (12i/π)(1/z)φ₋₂(z) - (36/π²)(1/z²)φ₋₄(z)

Note: The blueprint uses φ₋₂, φ₋₄ but Lean uses φ₂', φ₄' since negative subscripts are not valid identifiers.

Main Results

• φ₀_periodic: φ₀ is 1-periodic, i.e., φ₀(z + 1) = φ₀(z)
• φ₀_S_transform: S-transformation formula for φ₀

Supporting lemmas (in other files)

• E₂_periodic, E₂_S_transform: in E2.lean
• E₄_periodic, E₆_periodic, E₄_S_transform, E₆_S_transform: in Eisenstein.lean
• Δ_periodic, Δ_S_transform: in Delta.lean

Main Theorem: T-periodicity of φ₀

theorem φ₀_periodic (z : UpperHalfPlane) : φ₀ (1 +ᵥ z) = φ₀ z

φ₀ is 1-periodic: φ₀(z + 1) = φ₀(z)

Main Theorem: S-transformation of φ₀

theorem φ₀_S_transform (z : UpperHalfPlane) : φ₀ (ModularGroup.S • z) = φ₀ z - 12 * Complex.I / (↑Real.pi * ↑z) * φ₂' z - 36 / (↑Real.pi ^ 2 * ↑z ^ 2) * φ₄' z

The S-transformation formula for φ₀: φ₀(-1/z) = φ₀(z) - (12i/π)(1/z)φ₋₂(z) - (36/π²)(1/z²)φ₋₄(z)

This is Blueprint Lemma 7.2.