Congruence properties of the coefficients of the Drinfeld modular invariant (Q1865590)

The Fourier coefficients \(c(n)\) of the classical modular invariant \[ j = q^{-1}+744 + \sum_{n\geq 1} c(n)q^n \] are integers and satisfy remarkable congruences modulo powers of small primes. Replacing the ring \({\mathbb Z}\) by \(A = {\mathbb F}_q[T]\), the polynomial ring over a finite field \({\mathbb F}_q\), and elliptic curves, modular forms for \(\text{ SL}(2,{\mathbb Z})\), etc. by Drinfeld \(A\)-modules of rank two, modular forms for \(\text{ GL}(2,A)\), etc., the Drinfeld modular invariant \(j = j_{\text{ Drinfeld}}\) results. It has a similar expansion \[ j = -s^{-1} + \sum_{n\geq 0} c_ns^n \] (with the canonical uniformizer \(s\) at infinity adapted to the theory) with the coefficients \(c_n \in A\). Not much is known so far about these. The author manages, employing suitably modified ideas of J. Lehner stemming from the classical case, to prove congruences for the \(c_n\) modulo powers of ``small'' primes \(\mathfrak p\) of \(A\), where ``small'' means \(\deg \mathfrak p = 1\).