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.
Highlights
First we explain a general problem setting
We prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of Jacobi forms of general degree of any matrix index with respect to vector part arguments, are linear combinations of certain derivatives of vector valued Siegel modular forms of lower degrees obtained by the images of our differential operators
We show that our differential operators and related vectors of polynomials are obtained by a certain projection from these (Theorems 3.1, 3.2)
Summary
First we explain a general problem setting. Assume that Di are bounded symmetric domains for i = 1, 2 such that D2 ⊂ D1. Since we assumed that our differential operators D are linear and have constant coefficients, there are certain V -valued polynomials P(T ) over C in components of an n × n symmetric matrix T of variables such that. We prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of Jacobi forms of general degree of any matrix index with respect to vector part arguments, are linear combinations of certain derivatives of vector valued Siegel modular forms of lower degrees obtained by the images of our differential operators. We show that the original forms are recovered by these Siegel modular forms of lower degrees The proof of this part is not trivial at all and we need precise argument for existence of certain good operators. They are indispensable for explicit calculations of the critical values of the
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