Noncommutative Wilczynski Invariants, and Modular Differential Equations

Abstract

We develop a noncommutative invariant theory for ordinary linear differential operators on Riemann surfaces. For a monic binomially normalized operator $L=\sum_{k=0}^n {n\choose k}a_kD^{\,n-k}$, $a_0=1$, with coefficients in an associative differential algebra, we construct universal gauge-covariant coefficients $I_m(L)$. After correcting their reparametrization anomalies, we obtain Wilczy\'nski currents $W_m(L)$, which transform as genuine $m$-differentials. The construction is algebraic, finite-layered, and valid over noncommutative coefficient algebras; in the commutative scalar case it recovers the classical Wilczy\'nski invariants. We globalize the theory using jet bundles and infinitesimal neighborhoods of the diagonal. The natural global objects are $A$-linear opers, where $A$ is a sheaf of associative algebras with a compatible connection. In this setting $P=I_2/(n+1)$ is an $A_{\mathrm{ad}}$-valued projective connection, while $W_m$, $m\ge 3$, are global $A_{\mathrm{ad}}$-valued differentials; scalar invariants are obtained from traces, characteristic coefficients, and cyclic trace polynomials. As applications, we discuss projective connections, symmetric powers, fanning curves in Grassmannians, Calabi--Yau Picard--Fuchs equations, weak scalar and matrix-valued $W_2$-structures from Hodge subvariations, and modular differential equations. In the modular setting, the currents become modular forms, and the first coefficient gives the modular connection underlying the Serre derivative. We also extend the formalism to Siegel space using central Siegel modular connections and the associated equivariant differential algebra.