Invariants of formal pseudodifferential operator algebras and algebraic modular forms

Abstract

We study 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 conditions for such an extension to exist and describe under suitable assumptions the invariant subalgebras $B^\Gamma$ and $C^\Gamma$ as Laurent series rings with coefficients in $R^\Gamma$. We apply this general construction to the numbertheoretical context of a subgroup $\Gamma$ of $\mathrm{SL}(2,\mathbb{C})$ acting by homographies on an algebra $R$ of functions in one complex variable. The subalgebra $C_0^\Gamma$ of invariant operators of nonnegative order in $C^\Gamma$ is then linearly isomorphic to the product space $\mathcal{M}_0=\prod_{j\geq 0}M_j$, where $M_j$ is the vector space of algebraic modular forms of weight $j$ in $R$. We obtain a structure of noncommutative algebra on $\mathcal{M}_0$, which can be identified with a space of algebraic Jacobi forms. We study properties of the correspondence $\mathcal{M}_0\to C_0^\Gamma$, whose restriction to even weights was previously known, using arithmetical arguments and the algebraic results of the first part of the article.