Drinfeld Modular Forms Mirror Classical Hecke Eigenvalues and q-Expansion Arithmetic — E8 Intelligence Research

FINDING: Drinfeld modular forms connect Hecke eigenvalues to q-expansion coefficients over function fields, mirroring classical modular form arithmetic. | MATH: Classical: \(T_n f = \lambda_n f\), \(f(q)=\sum a_n q^n\), \(a_{mn}=a_m a_n\) for \((m,n)=1\), \(a_{p^{r+1}}=a_p a_{p^r}-p^{k-1}a_{p^{r-1}}\). Drinfeld analogue: Hecke operators \(T_{\mathfrak p}\) act on Drinfeld cusp forms; eigenvalues \(\lambda_{\mathfrak p}\) satisfy \(a_{\mathfrak p} = \lambda_{\mathfrak p}\) for good primes, with \(q\)-expansion over \(\mathbb F_q(T)\). Armana's work: bounds/relations for coefficients \(a_n\) in Drinfeld setting, including \(a_{n} = \lambda_{\mathfrak p} a_{n/\mathfrak p} - |\mathfrak p|^{k-1} a_{n/\mathfrak p^2}\) for \(\mathfrak p | n\). | CONNECTION: The Hecke algebra is a commutative ring generated by operators indexed by ideals — a lattice/root-system-like structure (the divisor lattice of \(\mathbb F_q[T]\)). Eigenvalues are algebraic integers, and the constant \(q^{k-1}\) (where \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com