Drinfeld Modular Forms as Function-Field Analogue of Classical Modular Forms — E8 Intelligence Research

FINDING: Drinfeld modular forms are the function-field analogue of classical modular forms, where the role of ℚ and its completions is replaced by a function field F_q(T) and its completions; Hecke eigenvalues and q-expansion coefficients obey analogous but structurally distinct arithmetic. | MATH: Classical modular forms: f(z) = Σ a_n q^n, q = e^{2πiz}, with Hecke operators T_p acting via a_{pn} = a_p a_n − p^{k−1} a_{n/p}. Drinfeld setting: base field K = F_q(T), ring A = F_q[T], Drinfeld upper half-plane Ω = ℙ¹(C_∞) − ℙ¹(K_∞) over a local field K_∞ ≅ F_q((1/T)). Drinfeld modular forms are functions on Ω with q-expansion f = Σ_{n≥0} a_n q^n where q = exp(2πi·(analogue of z)) in the rank-2 Drinfeld module setting; Hecke operators T_𝔭 indexed by primes 𝔭 ⊂ A satisfy a_{𝔭n} = a_𝔭 a_n − N(𝔭)^{k−1} a_{n/𝔭} with N(𝔭) = q^{deg 𝔭}. Gekeler's work establishes Eisenstein series, cusp forms, and the Drinfeld discriminant Δ with weight q−1 and q-expansion coefficients in F_q. Armana's results co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com