Slash action on E2 #

Overview #

This file proves how the weight-2 Eisenstein series E2 (and G2) transforms under the slash action of SL₂(ℤ). Unlike modular forms of weight k ≥ 4, the functions G2 and E2 are not modular forms because they acquire a correction term under the action of S = [[0, -1], [1, 0]].

Main Results #

G2_T_transform: G2 is invariant under T = [[1, 1], [0, 1]]: G₂|[2] T = G₂
G2_S_transform: The transformation under S involves a correction term: G₂(z) = z⁻² G₂(-1/z) - 2πi/z

Equivalently: G₂|[2] S = G₂ - D₂(S)
G2_slash_action: For any γ ∈ SL₂(ℤ): G₂|[2] γ = G₂ - D₂(γ)

where D₂(γ) is a correction term satisfying a cocycle condition.
E2_slash_action: The normalized version: E₂|[2] γ = E₂ - (1/(2ζ(2))) D₂(γ)

Proof Strategy #

The proof of G2_S_transform is the heart of this file. The strategy is:

Write as absolutely convergent series: Express G2 as an absolutely convergent double sum by adding and subtracting telescoping terms: G₂(z) = 2ζ(2) + ∑' m n, 1/((mz+n)²(mz+n+1)) + δ(m,n) where δ is a correction for boundary terms.
Swapping summation order: The S-action on G2 corresponds to swapping the roles of m and n. After writing as an abs. conv. series, we can legitimately swap the order because the series is absolutely convergent.
The correction term: The difference between the two orderings of summation is precisely the term -2πi/z, which comes from the results in the Summable file (tsum_symmetricIco_tsum_sub_eq vs tsum_tsum_symmetricIco_sub_eq).
General γ: Since SL₂(ℤ) is generated by S and T, and we've computed the transformation under both generators, we can deduce the transformation under any element using the cocycle property of D₂.