# Invariants of formal pseudodifferential operator algebras and algebraic modular forms

## TL;DR

This paper investigates the algebraic structure of formal pseudodifferential operator algebras, their invariants under group actions, and applies these findings to modular forms and related number-theoretic functions.

## Contribution

It provides necessary and sufficient conditions for extending group actions to pseudodifferential operator rings and establishes a novel connection between invariant operators and algebraic modular forms.

## Key findings

- Characterization of invariant subalgebras as Laurent series rings
- Construction of an isomorphism between invariant operators and modular forms
- Application of algebraic results to number theory and modular forms

## Abstract

We study from an algebraic point of view the question of extending an action of a group \(\Gamma\) on a commutative domain \(R\) to a formal pseudodifferential operator ring \(B=R(\!(x\,;\,d)\!)\) with coefficients in \(R\), as well as to some canonical quadratic extension \(C=R(\!(x^{1/2}\,;\,\frac 12 d)\!)_2\) of \(B\). We give a necessary and sufficient condition of compatibility between the action and the derivation $d$ of $R$ for such an extension to exist, and we determine all possible extensions of the action to \(B\) and \(C\). We describe under suitable assumptions the invariant subalgebras \(B^\Gamma\) and \(C^\Gamma\) as Laurent series rings with coefficients in \(R^\Gamma\). The main results of this general study are applied in a numbertheoretical context to the case where \(\Gamma\) is a subgroup of \({\rm SL}(2,\C)\) acting by homographies on an algebra \(R\) of functions in one complex variable. Denoting by \(M_j\) the vector space of algebraic modular forms in $R$ of weight \(j\) (even or odd), we build for any nonnegative integer \(k\) a linear isomorphism between the subspace \(C_k^\Gamma\) of invariant operators of order \(\geq k\) in \(C^\Gamma\) and the product space \(\mathcal{M}_k=\prod_{j\geq k}M_j\), which can be identified with a space of algebraic Jacobi forms of weight \(k\). It results in particular a structure of noncommutative algebra on \(\mathcal M_0\) and an algebra isomorphism \(\Psi:\mathcal M_0\to C_0^\Gamma\), whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.