Some formulas for the coefficients of Drinfeld modular forms (Q817735)

Let \(A=\mathbb F_q[T]\), \(K=\mathbb F_q(T)\), \(K_\infty\) the completion of \(K\) at \(1/T\) and \(C\) the completion of the algebraic closure of \(K_\infty\). The author considers sums of values of a specific sequence of Drinfeld modular functions \(j_n(z)\) for \(\Gamma= \text{GL}_2(A)\) on the upper half plane \(\Omega=C-K_\infty\). The \(j_n(z)\) are polynomials in \(j(z)\), the Drinfeld modular invariant, of degree \(n\), coefficients in \(A\) and leading coefficient \((-1)^n\). She finds some formulas for \(t\)-expansion coefficients of any Drinfeld modular form for \(\Gamma\). As an application she shows that the values of \(j(z)\) at points in the divisor of Drinfeld modular forms for \(\Gamma\) are algebraic over \(K\).