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Noncommutative Wilczynski Invariants, and Modular Differential Equations

Amir Jafari

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 $f\in K^\times$, the operator transforms by conjugation $L\mapsto f^{-1}Lf$. Using noncommutative complete Bell polynomials $P_m(u)$ and covariant Bell polynomials $Q_m(u)$ associated to the shifted derivation $Δ_{a_1}=D+\operatorname{ad}_{a_1}$, 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.

Noncommutative Wilczynski Invariants, and Modular Differential Equations

Abstract

We develop an explicit invariant calculus for monic -th order linear differential operators in the Ore algebra of a (possibly noncommutative) differential algebra : The formalism requires only the Leibniz rule for and extends to connection-type differentials into a -bimodule, so it applies in particular to matrix-valued meromorphic coefficients and to the -equivariant differential algebras that arise in automorphic settings. For a gauge change written unambiguously as with , the operator transforms by conjugation . Using noncommutative complete Bell polynomials and covariant Bell polynomials associated to the shifted derivation , we prove a closed Miura/oper expansion and we give universal explicit formulas for every gauge covariant in terms of -Bell polynomials. Assuming a central-jet chain rule for reparametrizations, we compute the full transformation laws of the and construct the projective (Wilczynski) covariants ; in particular we obtain explicit formulas for and a filtration-based construction scheme for the higher , together with explicit formulas for , , and . We globalize the theory to Riemann surfaces and holomorphic bundles, then formulate modular and Siegel modular differential operators via modular connections. In genus this yields noncommutative Rankin--Cohen brackets attached to -valued modular connections and their Maurer--Cartan realizations; in higher genus it yields -linear Siegel determinant brackets and ordered-determinant brackets with values in noncommutative coefficient algebras.
Paper Structure (136 sections, 55 theorems, 468 equations)

This paper contains 136 sections, 55 theorems, 468 equations.

Table of Contents

  1. Example 1: order $n=2$ and the Schwarzian.
  2. Example 2: order $n=3$ and the first $W$--current.
  3. Example 3: constant coefficients and Hilbert invariants.
  4. Example 4: matrix coefficients and trace invariants.
  5. Example 5: modular and Siegel modular settings.
  6. Projective differential geometry and Wilczyński invariants.
  7. Opers, Drinfeld--Sokolov reduction, and $W$--algebras.
  8. Noncommutative coefficients and invariant theory.
  9. Noncommutative Bell polynomials and Ore normal ordering.
  10. Modular and Siegel modular derivatives.
  11. Modular linear differential operators and Rankin--Cohen brackets.
  12. Hermitian symmetric domains, locally symmetric varieties, and automorphic bundles.
  13. Local theory: noncommutative gauge--Wilczynski invariants
  14. Noncommutative differential rings and Ore operators
  15. Differential rings
  16. ...and 121 more sections

Key Result

Proposition 2.1

Let $L\in\mathsf K\langle\mathsf D\rangle$ and consider the equation $L(y)=0$. If $y=f\,\tilde{y}$ for $f\in\mathsf K^\times$, then $\tilde{y}$ satisfies

Theorems & Definitions (208)

  • Definition 1.1: Noncommutative differential ring
  • Example 1.2: Matrix meromorphic functions
  • Definition 1.3: Ore extension $\mathsf K\langle\mathsf D\rangle$
  • Remark 1.4
  • Definition 1.5: First-order differential calculus
  • Example 1.6: Holomorphic differentials on a chart
  • Definition 1.7: Connections in the bimodule formalism
  • Remark 1.8: Differential rings as a special case
  • Remark 1.9: Connection coefficients
  • Definition 1.10: $\Gamma$-equivariant differential calculus
  • ...and 198 more