Noncommutative Wilczynski Invariants, and Modular Differential Equations

Abstract: We develop an explicit invariant calculus for monic $n$-th order linear differential operators in the Ore algebra of a (possibly noncommutative) differential algebra $(K,D)$: [ L \;=\; \sum_{i=0}^n \binom{n}{i}\,a_i\,D^{\,n-i}\qquad (a_0=1). ] The formalism requires only the Leibniz rule for $D$ and extends to connection-type differentials $d:K\toΩ$ into a $K$-bimodule, so it applies in particular to matrix-valued meromorphic coefficients and to the $Γ$-equivariant differential algebras that arise in automorphic settings. For a gauge change written unambiguously as $y=f\,\widetilde y$ with $f\in K<sup>\times$, the operator transforms by conjugation $L\mapsto f<sup>{-1}Lf$. Using noncommutative complete Bell polynomials $P_m(u)$ and covariant Bell polynomials $Q_m(u)$ associated to the shifted derivation $Δ<em>{a_1}=D+\operatorname{ad}</em>{a_1}$, we prove a closed Miura/oper expansion [ L\;=\;(D+a_1)^n+\binom{n}{2}I_2(D+a_1)^{n-2}+\cdots+I_n, ] and we give universal explicit formulas for every gauge covariant $I_k$ in terms of $Q$-Bell polynomials. Assuming a central-jet chain rule for reparametrizations, we compute the full transformation laws of the $I_k$ and construct the projective (Wilczynski) covariants $W_k$; in particular we obtain explicit formulas for $W_2,W_3,W_4$ and a filtration-based construction scheme for the higher $W_k$, together with explicit formulas for $W_4$, $W_5$, and $W_6$. We globalize the theory to Riemann surfaces and holomorphic bundles, then formulate modular and Siegel modular differential operators via modular connections. In genus $1$ this yields noncommutative Rankin--Cohen brackets attached to $\mathcal A$-valued modular connections and their Maurer--Cartan realizations; in higher genus it yields $g$-linear Siegel determinant brackets and ordered-determinant brackets with values in noncommutative coefficient algebras.