1994-02
articles
1994-02-01
2019-04-11T13:35
https://scigraph.springernature.com/explorer/license/
57-75
Modular forms and differential operators
http://link.springer.com/10.1007/BF02830874
research_article
en
false
In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ2 + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree).
Mathematical Sciences
Proceedings - Mathematical Sciences
2008-1359
2251-7456
Don
Zagier
pub.1002910991
dimensions_id
Pure Mathematics
doi
10.1007/bf02830874
Springer Nature - SN SciGraph project
Max Planck Institute for Mathematics
Max-Planck-Institut für Mathematik, Gottfried-Claren Str. 26, 53225, Bonn, Germany
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