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Journal Article

#### Generic differential operators on Siegel modular forms and special polynomials

##### External Ressource

https://doi.org/10.1007/s00029-020-00593-3

(Publisher version)

##### Fulltext (public)

Ibukiyama_Generic differential operators on Siegel modular forms and special polynomials_2020.pdf

(Publisher version), 605KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Ibukiyama, T. (2020). Generic differential operators on Siegel modular forms and special
polynomials.* Selecta Mathematica,* *26*(5): 66. doi:10.1007/s00029-020-00593-3.

Cite as: http://hdl.handle.net/21.11116/0000-0007-82C4-8

##### Abstract

Holomorphic vector valued differential operators acting on Siegel modular forms
and preserving automorphy under the restriction to diagonal blocks are important in
many respects, including application to critical values of L functions. Such differential
operators are associated with vectors of new special polynomials of several variables
defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular
forms. We will give formulas for all such polynomials in two different ways. One is to
describe them using polynomials characterized by monomials in off-diagonal block
variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We
construct an explicit generating series of polynomials mutually related under certain
mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential
operators of this sort are obtained by certain projections. This process exhausts all the
differential operators in question. This is also generic in the sense that for any number
of variables and block partitions, it is given by a recursive unified expression. As an
application, we prove that the Taylor coefficients of Siegel modular forms with respect
to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are
essentially vector valued Siegel modular forms of lower degrees, which are obtained
as images of the differential operators given above. We also show that the original
forms are recovered by the images of our operators. This is an ultimate generalization
of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results
and practical construction are also given.