Invariants of formal pseudodifferential operator algebras and algebraic modular forms
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François Dumas [1] ; François Martin [1]
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[1] Université Clermont Auvergne, Francia
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- Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 65, Nº. 1, 2023, págs. 1-31
- Idioma: inglés
- Enlaces
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Resumen
We study 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((x1/2;12d))2 of B. We give conditions for such an extension to exist and describe under suitable assumptions the invariant subalgebras BΓ and CΓ as Laurent series rings with coefficients in RΓ. We apply this general construction to the numbertheoretical context of a subgroup Γ of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. The subalgebra CΓ 0 of invariant operators of nonnegative order in CΓ is then linearly isomorphic to the product space M0 =Qj≥0 Mj , where Mj is the vector space of algebraic modular forms of weight j in R. We obtain a structure of noncommutative algebra on M0, which can be identified with a space of algebraic Jacobi forms. We study properties of the correspondence M0 → CΓ 0, whose restriction to even weights was previously known, using arithmetical arguments and the algebraic results of the first part of the article.
