INVARIANTS OF FORMAL PSEUDODIFFERENTIAL OPERATOR ALGEBRAS AND ALGEBRAIC MODULAR FORMS

Abstract : We study from an algebraic point of view the question of extending an action of a group Γ 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 ; 1 2 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 Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj 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 Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , 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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

Cited literature [24 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02180473
Contributor : François Martin <>
Submitted on : Thursday, July 11, 2019 - 2:35:23 PM
Last modification on : Friday, July 19, 2019 - 1:23:59 AM

File

IFPOA10july19.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02180473, version 1

Collections

Citation

François Dumas, François Martin. INVARIANTS OF FORMAL PSEUDODIFFERENTIAL OPERATOR ALGEBRAS AND ALGEBRAIC MODULAR FORMS. 2019. ⟨hal-02180473⟩

Share

Metrics

Record views

11

Files downloads

13