Modular Discriminant as the 24th Power of Dedekind Eta — E8 Intelligence Research

FINDING: The modular discriminant Δ(τ) is the 24th power of the Dedekind eta function, tying weight-12 modular forms to a weight-1/2 cusp form and to the j-invariant via the elliptic curve discriminant. MATH: - Dedekind eta: η(τ) = q^{1/24} ∏_{n≥1}(1 − q^n), q = e^{2πiτ}, τ ∈ upper half-plane. - Weight: η is a modular form of weight 1/2 (with multiplier system); η^{24} has weight 12. - Modular discriminant: Δ(τ) = η(τ)^{24} = (2π)^{12} · (g₂³ − 27g₃²) / (1728) up to normalization; equivalently Δ = g₂³ − 27g₃² (up to (2π)^{12} factor). - Eisenstein invariants: g₂(τ) = 60 G₄(τ), g₃(τ) = 140 G₆(τ), with G_{2k} the weight-2k Eisenstein series. - j-invariant: j(τ) = 1728 · g₂³ / (g₂³ − 27g₃²) = 1728 · g₂³ / Δ. - Fourier: Δ(τ) = (2π)^{12} Σ_{n≥1} τ(n) q^n, τ(n) = Ramanujan tau function, multiplicative, |τ(p)| ≤ 2p^{11/2} (Deligne). - Hardy–Ramanujan: partition function p(n) ~ (1/(4n√3)) exp(π√(2n/3)); the eta function's q-series generates p(n) via 1/η(τ) = q^{1/24} Σ p(n) q^n (Euler pentago Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com