Ramanujan's τ & Modular Forms
The Discriminant Δ, Eisenstein Series, and the Upper Half-Plane
|Δ(τ)| on upper half-plane
Δ(τ) = (2π)¹²η(τ)²⁴ = q∏(1−qⁿ)²⁴, q=e^{2πiτ}
τ(n) = coefficients: Δ = Σ τ(n)qⁿ. Ramanujan 1916: |τ(p)| ≤ 2p^{11/2} (proved by Deligne 1974!)
Modular forms live on the upper half-plane ℍ = {τ : Im(τ)>0}, invariant under SL(2,ℤ).