Modular j-Function & SL(2,Z) Symmetry

Phase portrait of j(τ) on the upper half-plane — fundamental domain of the modular group

τ = i
j(τ) = 1728
q = e^(2πiτ)
12
0.30
1.00
Modular j-function: j(τ) = E₄(τ)³/Δ(τ) = q⁻¹ + 744 + 196884q + 21493760q² + …, q = e²πiτ
SL(2,Z) invariance: j((aτ+b)/(cτ+d)) = j(τ) for all [[a,b],[c,d]] ∈ SL(2,Z). Generators: T: τ↦τ+1, S: τ↦−1/τ.
Special values: j(i) = 1728, j(e^{2πi/3}) = 0, j(i∞) = ∞.
Monstrous Moonshine: Coefficients 196884, 21493760, … are sums of dimensions of Monster group representations (McKay 1978, Borcherds 1992 Fields Medal).