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Invariants of Formal Pseudodifferential Operator Algebras and Algebraic Modular Forms by Dumas, François; Martin, François is a scholarly article available to read on EtoBox.

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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

Author

Dumas, François; Martin, François

Published

2019

Language

EN