Jacobi forms and differential operators: Odd weights

Differential Operators on Jacobi Forms of Several Variables

The theory of the classical Jacobi forms on H × C has been studied extensively by Eichler and Zagier[?]. Ziegler[?] developed a more general approach of Jacobi forms of higher degree. In [?] and [?], Gritsenko and Krieg studied Jacobi forms on H × Cn and showed that these kinds of Jacobi forms naturally arise in the Jacobi Fourier expansions of all kinds of automorphic forms in several variable...

Derivatives of Multivariate Bernstein Operators and Smoothness with Jacobi Weights

Differential Forms and Odd Symplectic Geometry

We remind the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. Ševera. We study relations of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformation for an odd symplectic form is a polynomial in matrix entries and a complete square. ...

Modular forms and differential operators

A~tract, In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for each n i> 0 a bilinear operation which assigns to two modular forms f and g of weight k and l a modular form If, g], of weight k + l + 2n. In the present paper we study these "Rankin-Cohen brackets" from t w o points of view. On the one hand we give ...

Geometry of Differential Operators, and Odd Laplace Operators

Let ∆ be an arbitrary linear differential operator of the second order acting on functions on a (super)manifold M . In local coordinates ∆ = 1 2 S ∂b∂a +T a ∂a +R. The principal symbol of ∆ is the symmetric tensor field S, or the quadratic function S = 1 2 Spbpa on T ∗M . The principal symbol can be understood as a symmetric “bracket” on functions: {f, g} := ∆(fg) − (∆f) g − (−1)f (∆g) + ∆(1) f...